Metamath Proof Explorer Most Recent Proofs |
||
Mirrors > Home > MPE Home > Th. List > Recent | ILE Most Recent Other > MM 100 |
The original proofs of theorems with recently shortened proofs can often be found by appending "OLD" to the theorem name, for example 19.43OLD for 19.43. The "OLD" versions are usually deleted after a year.
Other links Email: Norm Megill. Mailing list: Metamath Google Group Updated 21-Oct-2021 . Contributing: How can I contribute to Metamath? Syndication: RSS feed (courtesy of Dan Getz) Related wikis: Ghilbert site; Ghilbert Google Group.
Recent news items (7-Aug-2021) Version 0.198 of the metamath program fixes a bug in "write source ... /rewrap" that prevented end-of-sentence punctuation from appearing in column 79, causing some rewrapped lines to be shorter than necessary. Because this affects about 2000 lines in set.mm, you should use version 0.198 or later for rewrapping before submitting to GitHub.
(7-May-2021) Mario Carneiro has written a Metamath verifier in Lean.
(5-May-2021) Marnix Klooster has written a Metamath verifier in Zig.
(24-Mar-2021) Metamath was mentioned in a couple of articles about OpenAI: Researchers find that large language models struggle with math and What Is GPT-F?.
(26-Dec-2020) Version 0.194 of the metamath program adds the keyword "htmlexturl" to the $t comment to specify external versions of theorem pages. This keyward has been added to set.mm, and you must update your local copy of set.mm for "verify markup" to pass with the new program version.
(19-Dec-2020) Aleksandr A. Adamov has translated the Wikipedia Metamath page into Russian.
(19-Nov-2020) Eric Schmidt's checkmm.cpp was used as a test case for C'est, "a non-standard version of the C++20 standard library, with enhanced support for compile-time evaluation." See C++20 Compile-time Metamath Proof Verification using C'est.
(10-Nov-2020) Filip Cernatescu has updated the XPuzzle (Android app) to version 1.2. XPuzzle is a puzzle with math formulas derived from the Metamath system. At the bottom of the web page is a link to the Google Play Store, where the app can be found.
(7-Nov-2020) Richard Penner created a cross-reference guide between Frege's logic notation and the notation used by set.mm.
(4-Sep-2020) Version 0.192 of the metamath program adds the qualifier '/extract' to 'write source'. See 'help write source' and also this Google Group post.
(23-Aug-2020) Version 0.188 of the metamath program adds keywords Conclusion, Fact, Introduction, Paragraph, Scolia, Scolion, Subsection, and Table to bibliographic references. See 'help write bibliography' for the complete current list.
Color key: | Metamath Proof Explorer | Hilbert Space Explorer | User Mathboxes |
Date | Label | Description |
---|---|---|
Theorem | ||
25-Oct-2021 | cnv0 5454 | The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 4709, ax-nul 4717, ax-pr 4833. (Revised by KP, 25-Oct-2021.) |
⊢ ◡∅ = ∅ | ||
25-Oct-2021 | relopabi 5167 | A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 4709, ax-nul 4717, ax-pr 4833. (Revised by KP, 25-Oct-2021.) |
⊢ 𝐴 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ Rel 𝐴 | ||
25-Oct-2021 | ssrel 5130 | A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Remove dependency on ax-sep 4709, ax-nul 4717, ax-pr 4833. (Revised by KP, 25-Oct-2021.) |
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵))) | ||
24-Oct-2021 | uspgrunop 40416 | The union of two simple pseudographs (with the same vertex set): If 〈𝑉, 𝐸〉 and 〈𝑉, 𝐹〉 are simple pseudographs, then 〈𝑉, 𝐸 ∪ 𝐹〉 is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐺 ∈ USPGraph ) & ⊢ (𝜑 → 𝐻 ∈ USPGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) ⇒ ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UPGraph ) | ||
24-Oct-2021 | upgrunop 25785 | The union of two pseudographs (with the same vertex set): If 〈𝑉, 𝐸〉 and 〈𝑉, 𝐹〉 are pseudographs, then 〈𝑉, 𝐸 ∪ 𝐹〉 is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐺 ∈ UPGraph ) & ⊢ (𝜑 → 𝐻 ∈ UPGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) ⇒ ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UPGraph ) | ||
24-Oct-2021 | upgrun 25784 | The union 𝑈 of two pseudographs 𝐺 and 𝐻 with the same vertex set 𝑉 is a pseudograph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐺 ∈ UPGraph ) & ⊢ (𝜑 → 𝐻 ∈ UPGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) ⇒ ⊢ (𝜑 → 𝑈 ∈ UPGraph ) | ||
24-Oct-2021 | ushgrunop 25743 | The union of two (undirected) simple hypergraphs (with the same vertex set) represented as ordered pair: If 〈𝑉, 𝐸〉 and 〈𝑉, 𝐹〉 are simple hypergraphs, then 〈𝑉, 𝐸 ∪ 𝐹〉 is a (not necessarily simple) hypergraph - the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐺 ∈ USHGraph ) & ⊢ (𝜑 → 𝐻 ∈ USHGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) ⇒ ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UHGraph ) | ||
24-Oct-2021 | ushgrun 25742 | The union 𝑈 of two (undirected) simple hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a (not necessarily simple) hypergraph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐺 ∈ USHGraph ) & ⊢ (𝜑 → 𝐻 ∈ USHGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) ⇒ ⊢ (𝜑 → 𝑈 ∈ UHGraph ) | ||
24-Oct-2021 | uhgrunop 25741 | The union of two (undirected) hypergraphs (with the same vertex set) represented as ordered pair: If 〈𝑉, 𝐸〉 and 〈𝑉, 𝐹〉 are hypergraphs, then 〈𝑉, 𝐸 ∪ 𝐹〉 is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices). (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐺 ∈ UHGraph ) & ⊢ (𝜑 → 𝐻 ∈ UHGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) ⇒ ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UHGraph ) | ||
24-Oct-2021 | uhgrun 25740 | The union 𝑈 of two (undirected) hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a hypergraph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐺 ∈ UHGraph ) & ⊢ (𝜑 → 𝐻 ∈ UHGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) ⇒ ⊢ (𝜑 → 𝑈 ∈ UHGraph ) | ||
24-Oct-2021 | hashle00 13049 | If the size of a set is less than or equal to zero, the set must be empty. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Proof shortened by AV, 24-Oct-2021.) |
⊢ (𝑉 ∈ 𝑊 → ((#‘𝑉) ≤ 0 ↔ 𝑉 = ∅)) | ||
24-Oct-2021 | hashf 12987 | The size function maps all finite sets to their cardinality, as members of ℕ0, and infinite sets to +∞. TODO-AV: mark as OBSOLETE and replace it by hashfxnn0 12986? (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 13-Jul-2014.) (Proof shortened by AV, 24-Oct-2021.) |
⊢ #:V⟶(ℕ0 ∪ {+∞}) | ||
24-Oct-2021 | fun2d 5981 | The union of functions with disjoint domains is a function, deduction version of fun2 5980. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) | ||
23-Oct-2021 | prelpwi 4842 | A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.) (Proof shortened by AV, 23-Oct-2021.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶) | ||
22-Oct-2021 | zclmncvs 22756 | The set of integers (as a subring of itself) is a complex module, but not a complex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.) |
⊢ 𝑍 = (ringLMod‘ℤring) ⇒ ⊢ (𝑍 ∈ ℂMod ∧ 𝑍 ∉ ℂVec) | ||
22-Oct-2021 | qcvs 22755 | The set of rational numbers (as a subring of itself) is a complex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.) |
⊢ 𝑄 = (ringLMod‘(ℂfld ↾s ℚ)) ⇒ ⊢ 𝑄 ∈ ℂVec | ||
22-Oct-2021 | recvs 22754 | The set of real numbers (as a subring of itself) is a complex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.) |
⊢ 𝑅 = (ringLMod‘ℝfld) ⇒ ⊢ 𝑅 ∈ ℂVec | ||
22-Oct-2021 | zringndrg 19657 | The integers are not a division ring, and therefore not a field. (Contributed by AV, 22-Oct-2021.) |
⊢ ℤring ∉ DivRing | ||
21-Oct-2021 | nfsetrecs 42232 | Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.) |
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ Ⅎ𝑥setrecs(𝐹) | ||
20-Oct-2021 | bj-axc14 32032 | Alternate proof of axc14 2360 (even when inlining the above results, this gives a shorter proof). (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))) | ||
20-Oct-2021 | bj-axc14nf 32031 | Proof of a version of axc14 2360 using the "non-free" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥 ∈ 𝑦)) | ||
20-Oct-2021 | bj-nfeel2 32030 | Non-freeness in an equality. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧) | ||
20-Oct-2021 | bj-dvelimv 32029 | A version of dvelim 2325 using the "non-free" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜑)) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑) | ||
20-Oct-2021 | bj-dvelimdv1 32028 | Curried form (exported form) of bj-dvelimdv 32027. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓)) ⇒ ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)) | ||
20-Oct-2021 | bj-dvelimdv 32027 |
Deduction form of dvelim 2325 with DV conditions. Typically, 𝑧 is a
fresh variable used for the implicit substitution hypothesis that
results in 𝜒 (namely, 𝜓 can be thought as 𝜓(𝑥, 𝑦) and
𝜒 as 𝜓(𝑥, 𝑧)). So the theorem says that if x is
effectively free in 𝜓(𝑥, 𝑧), then if x and y are not the same
variable, then 𝑥 is also effectively free in 𝜓(𝑥, 𝑦), in a
context 𝜑.
One can weakend the implicit substitution hypothesis by adding the antecedent 𝜑 but this typically does not make the theorem much more useful. Similarly, one could use non-freeness hypotheses instead of DV conditions but since this result is typically used when 𝑧 is a dummy variable, this would not be of much benefit. One could also remove DV(z,x) since in the proof nfv 1830 can be replaced with nfal 2139 followed by nfn 1768. Remark: nfald 2151 uses ax-11 2021; it might be possible to inline and use ax11w 1994 instead, but there is still a use via 19.12 2150 anyway. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓)) ⇒ ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | ||
20-Oct-2021 | bj-nfimt2 32026 | Uncurried form of bj-nfimt 32025 and closed form of nfim 1813. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ ((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜓) → Ⅎ𝑥(𝜑 → 𝜓)) | ||
20-Oct-2021 | bj-nfimt 32025 | Closed form of nfim 1813. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓))) | ||
20-Oct-2021 | bj-nfbiit 32024 | Closed form of nfbii 1770 (the label " nfbi 1821 " is taken for another result. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)) | ||
20-Oct-2021 | bj-syl66ib 32023 | A mixed syllogism inference derived from syl6ib 240. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜃 → 𝜏) & ⊢ (𝜏 ↔ 𝜒) ⇒ ⊢ (𝜑 → (𝜓 → 𝜒)) | ||
20-Oct-2021 | syl2imc 40 | A commuted version of syl2im 39. Implication-only version of syl2anr 494. (Contributed by BJ, 20-Oct-2021.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜃) & ⊢ (𝜓 → (𝜃 → 𝜏)) ⇒ ⊢ (𝜒 → (𝜑 → 𝜏)) | ||
19-Oct-2021 | cphipipcj 22808 | An inner product times its conjugate. (Contributed by NM, 23-Nov-2007.) (Revised by AV, 19-Oct-2021.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 , 𝐵) · (𝐵 , 𝐴)) = ((abs‘(𝐴 , 𝐵))↑2)) | ||
19-Oct-2021 | tngngpim 22273 | The induced metric of a normed group is a function. (Contributed by AV, 19-Oct-2021.) |
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐷 = (dist‘𝑇) ⇒ ⊢ (𝐺 ∈ NrmGrp → 𝐷:(𝑋 × 𝑋)⟶ℝ) | ||
19-Oct-2021 | sgrimval 22246 | The induced metric on a subgroup in terms of the induced metric on the parent normed group. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 19-Oct-2021.) |
⊢ 𝑋 = (𝑇 ↾s 𝑈) & ⊢ 𝐷 = (dist‘𝑇) & ⊢ 𝐸 = (dist‘𝑋) & ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ 𝑆 = (SubGrp‘𝑇) ⇒ ⊢ (((𝐺 ∈ NrmGrp ∧ 𝑈 ∈ 𝑆) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈)) → (𝐴𝐸𝐵) = (𝐴𝐷𝐵)) | ||
19-Oct-2021 | sgrim 22245 | The induced metric on a subgroup is the induced metric on the parent group equipped with a norm. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 19-Oct-2021.) |
⊢ 𝑋 = (𝑇 ↾s 𝑈) & ⊢ 𝐷 = (dist‘𝑇) & ⊢ 𝐸 = (dist‘𝑋) ⇒ ⊢ (𝑈 ∈ 𝑆 → 𝐸 = 𝐷) | ||
19-Oct-2021 | phssip 19822 | The inner product (as a function) on a subspace is a restriction of the inner product (as a function) on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ · = (·if‘𝑊) & ⊢ 𝑃 = (·if‘𝑋) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑃 = ( · ↾ (𝑈 × 𝑈))) | ||
19-Oct-2021 | phssipval 19821 | The inner product on a subspace in terms of the inner product on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑃 = (·𝑖‘𝑋) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈)) → (𝐴𝑃𝐵) = (𝐴 , 𝐵)) | ||
19-Oct-2021 | ssipeq 19820 | The inner product on a subspace equals the inner product on the parent space. (Contributed by AV, 19-Oct-2021.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑃 = (·𝑖‘𝑋) ⇒ ⊢ (𝑈 ∈ 𝑆 → 𝑃 = , ) | ||
19-Oct-2021 | oprres 6700 | The restriction of an operation is an operation. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 19-Oct-2021.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) & ⊢ (𝜑 → 𝑌 ⊆ 𝑋) & ⊢ (𝜑 → 𝐹:(𝑌 × 𝑌)⟶𝑅) & ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)⟶𝑆) ⇒ ⊢ (𝜑 → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌))) | ||
18-Oct-2021 | cphipval 22850 | Value of the inner product expressed by a sum of terms with the norm defined by the inner product. Equation 6.45 of [Ponnusamy] p. 361. (Contributed by NM, 31-Jan-2007.) (Revised by AV, 18-Oct-2021.) |
⊢ 𝑋 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴 + ((i↑𝑘) · 𝐵)))↑2)) / 4)) | ||
18-Oct-2021 | 4cphipval2 22849 | Four times the inner product value cphipval2 22848. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 18-Oct-2021.) |
⊢ 𝑋 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (4 · (𝐴 , 𝐵)) = ((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2))))) | ||
18-Oct-2021 | cphipval2 22848 | Value of the inner product expressed by the norm defined by it. (Contributed by NM, 31-Jan-2007.) (Revised by AV, 18-Oct-2021.) |
⊢ 𝑋 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐵) = (((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)))) / 4)) | ||
18-Oct-2021 | cmodscmulexp 22730 | The scalar product of a vector with powers of i belongs to the base set of a complex module if the scalar subring of th complex module contains i. (Contributed by AV, 18-Oct-2021.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑋 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋 ∧ 𝑁 ∈ ℕ)) → ((i↑𝑁) · 𝐵) ∈ 𝑋) | ||
18-Oct-2021 | cmodscexp 22729 | The powers of i belong to the scalar subring of a complex module if i belongs to the scalar subring . (Contributed by AV, 18-Oct-2021.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (((𝑊 ∈ ℂMod ∧ i ∈ 𝐾) ∧ 𝑁 ∈ ℕ) → (i↑𝑁) ∈ 𝐾) | ||
18-Oct-2021 | addneg1mul 10351 | Addition with product with minus one is a subtraction. (Contributed by AV, 18-Oct-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (-1 · 𝐵)) = (𝐴 − 𝐵)) | ||
18-Oct-2021 | nfa2 2027 | Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2034. (Revised by Wolf Lammen, 18-Oct-2021.) |
⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑 | ||
17-Oct-2021 | cphassir 22823 | "Associative" law for the second argument of an inner product with scalar _𝑖. (Contributed by AV, 17-Oct-2021.) |
⊢ 𝑋 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , (i · 𝐵)) = (-i · (𝐴 , 𝐵))) | ||
17-Oct-2021 | cphassi 22822 | Associative law for the first argument of an inner product with scalar _𝑖. (Contributed by AV, 17-Oct-2021.) |
⊢ 𝑋 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · 𝐵) , 𝐴) = (i · (𝐵 , 𝐴))) | ||
17-Oct-2021 | cnindmet 22770 | The metric induced on the complex numbers. cnmet 22385 proves that it is a metric. The induced metric is identical with the original metric on the complex numbers, see cnfldds 19577 and also cnmet 22385. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by AV, 17-Oct-2021.) |
⊢ 𝑇 = (ℂfld toNrmGrp abs) ⇒ ⊢ (dist‘𝑇) = (abs ∘ − ) | ||
17-Oct-2021 | ngnmcncn 22456 | The norm of a normed group is a continuous function to ℂ. (Contributed by NM, 12-Aug-2007.) (Revised by AV, 17-Oct-2021.) |
⊢ 𝑁 = (norm‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ (𝐺 ∈ NrmGrp → 𝑁 ∈ (𝐽 Cn 𝐾)) | ||
16-Oct-2021 | nglmle 22908 | If the norm of each member of a converging sequence is less than or equal to a given amount, so is the norm of the convergence value. (Contributed by NM, 25-Dec-2007.) (Revised by AV, 16-Oct-2021.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋)) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ NrmGrp) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑁‘(𝐹‘𝑘)) ≤ 𝑅) ⇒ ⊢ (𝜑 → (𝑁‘𝑃) ≤ 𝑅) | ||
16-Oct-2021 | tngngp3 22270 | Alternate definition of a normed group (i.e. a group equipped with a norm) without using the properties of a metric space. This corresponds to the definition in N. H. Bingham, A. J. Ostaszewski: "Normed versus topological groups: dichotomy and duality", 2010, Dissertationes Mathematicae 472, pp. 1-138 and E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006. (Contributed by AV, 16-Oct-2021.) |
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))) | ||
16-Oct-2021 | nfald 2151 | Deduction form of nfal 2139. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) (Proof shortened by Wolf Lammen, 16-Oct-2021.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) | ||
16-Oct-2021 | hbex 2142 | If 𝑥 is not free in 𝜑, it is not free in ∃𝑦𝜑. (Contributed by NM, 12-Mar-1993.) Reduce symbol count in nfex 2140, hbex 2142. (Revised by Wolf Lammen, 16-Oct-2021.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) | ||
16-Oct-2021 | nfex 2140 | If 𝑥 is not free in 𝜑, it is not free in ∃𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Reduce symbol count in nfex 2140, hbex 2142. (Revised by Wolf Lammen, 16-Oct-2021.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃𝑦𝜑 | ||
15-Oct-2021 | nrmtngnrm 22272 | The augmentation of a normed group by its own norm is a normed group with the same norm. (Contributed by AV, 15-Oct-2021.) |
⊢ 𝑇 = (𝐺 toNrmGrp (norm‘𝐺)) ⇒ ⊢ (𝐺 ∈ NrmGrp → (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) = (norm‘𝐺))) | ||
15-Oct-2021 | nrmtngdist 22271 | The augmentation of a normed group by its own norm has the same distance function as the normed group (restricted to the base set). (Contributed by AV, 15-Oct-2021.) |
⊢ 𝑇 = (𝐺 toNrmGrp (norm‘𝐺)) & ⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ NrmGrp → (dist‘𝑇) = ((dist‘𝐺) ↾ (𝑋 × 𝑋))) | ||
15-Oct-2021 | tnggrpr 22269 | If a structure equipped with a norm is a normed group, the structure itself must be a group. (Contributed by AV, 15-Oct-2021.) |
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) ⇒ ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp) | ||
15-Oct-2021 | distspace 21931 | A structure 𝐺 with a distance function 𝐷 which is a pseudometric is a distance space (according to E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006), i.e. a (base) set 𝑋 equipped with a distance 𝐷, which is a mapping of two elements of the base set to the (extended) reals and which is non-negative, symmetric and equal to 0 if the two elements are equal. (Contributed by AV, 15-Oct-2021.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ (𝐴𝐷𝐴) = 0) ∧ (0 ≤ (𝐴𝐷𝐵) ∧ (𝐴𝐷𝐵) = (𝐵𝐷𝐴)))) | ||
15-Oct-2021 | nfrd 1708 | Consequence of the definition of not-free in a context. (Contributed by Wolf Lammen, 15-Oct-2021.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) | ||
14-Oct-2021 | ngpocelbl 22318 | Membership of an off-center vector in a ball in a normed module. (Contributed by NM, 27-Dec-2007.) (Revised by AV, 14-Oct-2021.) |
⊢ 𝑁 = (norm‘𝐺) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋)) ⇒ ⊢ ((𝐺 ∈ NrmMod ∧ 𝑅 ∈ ℝ* ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑃 + 𝐴) ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑁‘𝐴) < 𝑅)) | ||
14-Oct-2021 | ngpmet 22217 | The (induced) metric of a normed group is a metric. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 14-Oct-2021.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝐺 ∈ NrmGrp → 𝐷 ∈ (Met‘𝑋)) | ||
14-Oct-2021 | hbnt 2129 | Closed theorem version of bound-variable hypothesis builder hbn 2131. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) (Proof shortened by Wolf Lammen, 14-Oct-2021.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | ||
13-Oct-2021 | ncvspds 22769 | Value of the distance function in terms of the norm of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (Revised by AV, 13-Oct-2021.) |
⊢ 𝑁 = (norm‘𝐺) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) & ⊢ · = ( ·𝑠 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 + (-1 · 𝐵)))) | ||
12-Oct-2021 | hba1 2137 | The setvar 𝑥 is not free in ∀𝑥𝜑. This corresponds to the axiom (4) of modal logic. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 12-Oct-2021.) |
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | ||
12-Oct-2021 | axc16nf 2122 | If dtru 4783 is false, then there is only one element in the universe, so everything satisfies Ⅎ. (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2021. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2006. (Revised by Wolf lammen, 12-Oct-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) | ||
12-Oct-2021 | nfnf1 2018 | The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2034. (Revised by Wolf Lammen, 12-Oct-2021.) |
⊢ Ⅎ𝑥Ⅎ𝑥𝜑 | ||
12-Oct-2021 | nfa1 2015 | The setvar 𝑥 is not free in ∀𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 11-Sep-2021.) Remove dependency on ax-12 2034. (Revised by Wolf Lammen, 12-Oct-2021.) |
⊢ Ⅎ𝑥∀𝑥𝜑 | ||
11-Oct-2021 | axc16gOLD 2147 | Obsolete proof of axc16g 2119 as of 11-Oct-2021. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) Remove dependency on ax-13 2234, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2124. (Revised by Wolf Lammen, 11-Oct-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | ||
11-Oct-2021 | axc11rv 2124 | Version of axc11r 2175 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 1922 }, from ax12v 2035 (contrary to axc11 2302 which seems to require the full ax-12 2034). (Contributed by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑)) | ||
11-Oct-2021 | axc11v 2123 | Version of axc11 2302 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 1922 }, from ax12v 2035 (contrary to axc11 2302 which seems to require the full ax-12 2034 and ax-13 2234). (Contributed by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
11-Oct-2021 | axc16g 2119 | Generalization of axc16 2120. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 1922 }, theorem ax12v 2035. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) Remove dependency on ax-13 2234, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2124. (Revised by Wolf Lammen, 11-Oct-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | ||
11-Oct-2021 | stdpc5 2063 | An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis Ⅎ𝑥𝜑 can be thought of as emulating "𝑥 is not free in 𝜑." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example 𝑥 would not (for us) be free in 𝑥 = 𝑥 by nfequid 1927. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. See stdpc5v 1854 for a version requiring fewer axioms. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 4-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) | ||
11-Oct-2021 | nf5dh 2013 | Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 11-Oct-2021.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
10-Oct-2021 | gtinf 31483 | Any number greater than an infimum is greater than some element of the set. (Contributed by Jeff Hankins, 29-Sep-2013.) (Revised by AV, 10-Oct-2021.) |
⊢ (((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) ∧ (𝐴 ∈ ℝ ∧ inf(𝑆, ℝ, < ) < 𝐴)) → ∃𝑧 ∈ 𝑆 𝑧 < 𝐴) | ||
10-Oct-2021 | wsuclb 31021 | A well-founded successor is a lower bound on points after 𝑋. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝑋𝑅𝑌) ⇒ ⊢ (𝜑 → ¬ 𝑌𝑅wsuc(𝑅, 𝐴, 𝑋)) | ||
10-Oct-2021 | wsuccl 31020 | If 𝑋 is a set with an 𝑅 successor in 𝐴, then its well-founded successor is a member of 𝐴. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋𝑅𝑦) ⇒ ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ 𝐴) | ||
10-Oct-2021 | wsucex 31019 | Existence theorem for well-founded successor. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → wsuc(𝑅, 𝐴, 𝑋) ∈ V) | ||
10-Oct-2021 | wsuclem 31017 | Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦))) | ||
10-Oct-2021 | wzel 31015 | The zero of a well-founded set is a member of that set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ 𝐴 ≠ ∅) → inf(𝐴, 𝐴, 𝑅) ∈ 𝐴) | ||
10-Oct-2021 | elwlim 31013 | Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
⊢ (𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅))) | ||
10-Oct-2021 | nfwlim 31012 | Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥WLim(𝑅, 𝐴) | ||
10-Oct-2021 | wlimeq12 31009 | Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵)) | ||
10-Oct-2021 | nfwsuc 31008 | Bound-variable hypothesis builder for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝑋 ⇒ ⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋) | ||
10-Oct-2021 | wsuceq123 31004 | Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌)) | ||
10-Oct-2021 | df-wlim 31002 | Define the class of limit points of a well-founded set. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))} | ||
10-Oct-2021 | df-wsuc 31000 | Define the concept of a successor in a well-founded set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | ||
10-Oct-2021 | inffz 30867 | The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by AV, 10-Oct-2021.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀) | ||
10-Oct-2021 | df-qp 30807 | Define the 𝑝-adic completion of the rational numbers, as a normed field structure with a total order (that is not compatible with the operations). (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 10-Oct-2021.) |
⊢ Qp = (𝑝 ∈ ℙ ↦ ⦋{ℎ ∈ (ℤ ↑𝑚 (0...(𝑝 − 1))) ∣ ∃𝑥 ∈ ran ℤ≥(◡ℎ “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑏⦌(({〈(Base‘ndx), 𝑏〉, 〈(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑓 ∘𝑓 + 𝑔)))〉, 〈(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑛 ∈ ℤ ↦ Σ𝑘 ∈ ℤ ((𝑓‘𝑘) · (𝑔‘(𝑛 − 𝑘))))))〉} ∪ {〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑏 ∧ Σ𝑘 ∈ ℤ ((𝑓‘-𝑘) · ((𝑝 + 1)↑-𝑘)) < Σ𝑘 ∈ ℤ ((𝑔‘-𝑘) · ((𝑝 + 1)↑-𝑘)))}〉}) toNrmGrp (𝑓 ∈ 𝑏 ↦ if(𝑓 = (ℤ × {0}), 0, (𝑝↑-inf((◡𝑓 “ (ℤ ∖ {0})), ℝ, < )))))) | ||
10-Oct-2021 | ax13dgen4 2004 | Degenerate instance of ax-13 2234 where bundled variables 𝑥, 𝑦, and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Oct-2021.) |
⊢ (¬ 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) | ||
10-Oct-2021 | hba1w 1961 | Weak version of hba1 2137. See comments for ax10w 1993. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | ||
10-Oct-2021 | spfw 1952 | Weak version of sp 2041. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.) |
⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) & ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) & ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜑) | ||
10-Oct-2021 | nexdv 1851 | Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 13-Jul-2020.) (Proof shortened by Wolf Lammen, 10-Oct-2021.) |
⊢ (𝜑 → ¬ 𝜓) ⇒ ⊢ (𝜑 → ¬ ∃𝑥𝜓) | ||
9-Oct-2021 | cnncvsabsnegdemo 22773 | Derive the absolute value of a negative complex number absneg 13865 to demonstrate the use of the properties of a normed complex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴)) | ||
9-Oct-2021 | cnncvsmulassdemo 22772 | Derive the associative law for complex number multiplication mulass 9903 interpreted as scalar multiplication to demonstrate the use of the properties of a (normed) complex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) | ||
9-Oct-2021 | cnncvsaddassdemo 22771 | Derive the associative law for complex number addition addass 9902 to demonstrate the use of the properties of a (normed complex) vector space for the complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by AV, 9-Oct-2021.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) | ||
9-Oct-2021 | cnnm 22768 | The norm operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by AV, 9-Oct-2021.) |
⊢ 𝐶 = (ringLMod‘ℂfld) ⇒ ⊢ (norm‘𝐶) = abs | ||
9-Oct-2021 | cnncvs 22767 | The set of complex numbers (as a subring of itself) is a normed complex vector space. The vector operation is +, and the scalar product is ·, and the norm function is abs. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 9-Oct-2021.) |
⊢ 𝐶 = (ringLMod‘ℂfld) ⇒ ⊢ 𝐶 ∈ (NrmVec ∩ ℂVec) | ||
9-Oct-2021 | cnrnvc 22766 | The set of complex numbers (as a subring of itself) is a normed vector space over itself. The vector operation is +, and the scalar product is ·, and the norm function is abs. (Contributed by AV, 9-Oct-2021.) |
⊢ 𝐶 = (ringLMod‘ℂfld) ⇒ ⊢ 𝐶 ∈ NrmVec | ||
9-Oct-2021 | rlmnm 22303 | The norm function in the ring module. (Contributed by AV, 9-Oct-2021.) |
⊢ (norm‘𝑅) = (norm‘(ringLMod‘𝑅)) | ||
9-Oct-2021 | nfan 1816 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 9-Oct-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) | ||
8-Oct-2021 | ncvs1 22765 | From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (Revised by AV, 8-Oct-2021.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ · = ( ·𝑠 ‘𝐺) & ⊢ 𝐹 = (Scalar‘𝐺) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 0 ) ∧ (1 / (𝑁‘𝐴)) ∈ 𝐾) → (𝑁‘((1 / (𝑁‘𝐴)) · 𝐴)) = 1) | ||
8-Oct-2021 | ncvspi 22764 | The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (Revised by AV, 8-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑁‘(𝐴 + (i · 𝐵))) = (𝑁‘(𝐵 + (-i · 𝐴)))) | ||
8-Oct-2021 | ncvsdif 22763 | The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (Revised by AV, 8-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴 + (-1 · 𝐵))) = (𝑁‘(𝐵 + (-1 · 𝐴)))) | ||
8-Oct-2021 | ncvsm1 22762 | The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (Revised by AV, 8-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = (𝑁‘𝐴)) | ||
8-Oct-2021 | ncvsge0 22761 | The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (Revised by AV, 8-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ (𝐾 ∩ ℝ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴 · 𝐵)) = (𝐴 · (𝑁‘𝐵))) | ||
8-Oct-2021 | ncvsprp 22760 | Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by AV, 8-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) | ||
8-Oct-2021 | nmgt0 22244 | The norm of a nonzero element is a positive real. (Contributed by NM, 20-Nov-2007.) (Revised by AV, 8-Oct-2021.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝐴 ≠ 0 ↔ 0 < (𝑁‘𝐴))) | ||
8-Oct-2021 | nmtri2 22241 | Triangle inequality for the norm of two subtractions. (Contributed by NM, 24-Feb-2008.) (Revised by AV, 8-Oct-2021.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝑁‘(𝐴 − 𝐶)) ≤ ((𝑁‘(𝐴 − 𝐵)) + (𝑁‘(𝐵 − 𝐶)))) | ||
8-Oct-2021 | exan 1775 | Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) Reduce axiom dependencies. (Revised by BJ, 7-Jul-2021.) (Proof shortened by Wolf Lammen, 8-Oct-2021.) |
⊢ (∃𝑥𝜑 ∧ 𝜓) ⇒ ⊢ ∃𝑥(𝜑 ∧ 𝜓) | ||
7-Oct-2021 | ncvsi 22759 | The properties of a normed complex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → (𝑊 ∈ ℂVec ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))))) | ||
7-Oct-2021 | isncvsngpd 22758 | Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (Revised by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ ℂVec) & ⊢ (𝜑 → 𝑊 ∈ NrmGrp) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑘 ∈ 𝐾)) → (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) ⇒ ⊢ (𝜑 → 𝑊 ∈ (NrmVec ∩ ℂVec)) | ||
7-Oct-2021 | isncvsngp 22757 | The predicate "is a normed complex vector space." if the underlying group is a normed group. (Contributed by NM, 5-Jun-2008.) (Revised by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)))) | ||
7-Oct-2021 | clmvz 22719 | Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (Revised by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ( 0 − 𝐴) = (-1 · 𝐴)) | ||
7-Oct-2021 | clmvsubval2 22718 | Value of vector subtraction on a complex module. (Contributed by Mario Carneiro, 19-Nov-2013.) (Revised by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = ((-1 · 𝐵) + 𝐴)) | ||
7-Oct-2021 | clmvsubval 22717 | Value of vector subtraction in terms of addition in a complex module. (lmodvsubval2 18741 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + (-1 · 𝐵))) | ||
7-Oct-2021 | clmvscom 22698 | Commutative law for the scalar product. (Contributed by NM, 14-Feb-2008.) (Revised by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝑄 · (𝑅 · 𝑋)) = (𝑅 · (𝑄 · 𝑋))) | ||
7-Oct-2021 | ngpi 22242 | The properties of a normed group, which is a group accompanied by a norm. (Contributed by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ (𝑊 ∈ NrmGrp → (𝑊 ∈ Grp ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) | ||
7-Oct-2021 | ablsubsub23 18053 | Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007.) (Revised by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Abel ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐴 − 𝐶) = 𝐵)) | ||
4-Oct-2021 | iscvsi 22737 | Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 ∈ Grp & ⊢ 𝑆 = (ℂfld ↾s 𝐾) & ⊢ 𝑆 ∈ DivRing & ⊢ 𝐾 ∈ (SubRing‘ℂfld) & ⊢ (𝑥 ∈ 𝑉 → (1 · 𝑥) = 𝑥) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑦 · 𝑥) ∈ 𝑉) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧))) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥))) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))) ⇒ ⊢ 𝑊 ∈ ℂVec | ||
4-Oct-2021 | iscvsp 22736 | The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) ⇒ ⊢ (𝑊 ∈ ℂVec ↔ ((𝑊 ∈ Grp ∧ (𝑆 ∈ DivRing ∧ 𝑆 = (ℂfld ↾s 𝐾)) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥 ∈ 𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))))) | ||
4-Oct-2021 | iscvs 22735 | A complex vector space is a complex left module over a division ring. For example, the (complex) left modules over the rational or real or complex numbers are complex vector spaces. (Contributed by AV, 4-Oct-2021.) |
⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing)) | ||
4-Oct-2021 | isclmi0 22706 | Properties that determine a complex left module. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑆 = (ℂfld ↾s 𝐾) & ⊢ 𝑊 ∈ Grp & ⊢ 𝐾 ∈ (SubRing‘ℂfld) & ⊢ (𝑥 ∈ 𝑉 → (1 · 𝑥) = 𝑥) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑦 · 𝑥) ∈ 𝑉) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧))) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥))) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))) ⇒ ⊢ 𝑊 ∈ ℂMod | ||
4-Oct-2021 | isclmp 22705 | The predicate "is a complex left module space." (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) ⇒ ⊢ (𝑊 ∈ ℂMod ↔ ((𝑊 ∈ Grp ∧ 𝑆 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥 ∈ 𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))))) | ||
4-Oct-2021 | nfnt 1767 | If 𝑥 is not free in 𝜑, then it is not free in ¬ 𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.) df-nf 1701 changed. (Revised by Wolf Lammen, 4-Oct-2021.) |
⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑) | ||
3-Oct-2021 | clmopfne 22704 | The (functionalized) operations of a complex left module cannot be identical. (Contributed by NM, 31-May-2008.) (Revised by AV, 3-Oct-2021.) |
⊢ · = ( ·sf ‘𝑊) & ⊢ + = (+𝑓‘𝑊) ⇒ ⊢ (𝑊 ∈ ℂMod → + ≠ · ) | ||
2-Oct-2021 | lmodfopne 18724 | The (functionalized) operations of a left module (over a nonzero ring) cannot be identical. (Contributed by NM, 31-May-2008.) (Revised by AV, 2-Oct-2021.) |
⊢ · = ( ·sf ‘𝑊) & ⊢ + = (+𝑓‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) & ⊢ 1 = (1r‘𝑆) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 1 ≠ 0 ) → + ≠ · ) | ||
2-Oct-2021 | lmodfopnelem2 18723 | Lemma 2 for lmodfopne 18724. (Contributed by AV, 2-Oct-2021.) |
⊢ · = ( ·sf ‘𝑊) & ⊢ + = (+𝑓‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) & ⊢ 1 = (1r‘𝑆) ⇒ ⊢ ((𝑊 ∈ LMod ∧ + = · ) → ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉)) | ||
2-Oct-2021 | lmodfopnelem1 18722 | Lemma 1 for lmodfopne 18724. (Contributed by AV, 2-Oct-2021.) |
⊢ · = ( ·sf ‘𝑊) & ⊢ + = (+𝑓‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) ⇒ ⊢ ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾) | ||
29-Sep-2021 | nmoleub3 22727 | The operator norm is the supremum of the value of a linear operator on the closed unit sphere. (Contributed by Mario Carneiro, 19-Oct-2015.) (Proof shortened by AV, 29-Sep-2021.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 𝐿 = (norm‘𝑆) & ⊢ 𝑀 = (norm‘𝑇) & ⊢ 𝐺 = (Scalar‘𝑆) & ⊢ 𝐾 = (Base‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → ℝ ⊆ 𝐾) ⇒ ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))) | ||
29-Sep-2021 | nmoleub2lem3 22723 | Lemma for nmoleub2a 22725 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.) (Proof shortened by AV, 29-Sep-2021.) |
⊢ 𝑁 = (𝑆 normOp 𝑇) & ⊢ 𝑉 = (Base‘𝑆) & ⊢ 𝐿 = (norm‘𝑆) & ⊢ 𝑀 = (norm‘𝑇) & ⊢ 𝐺 = (Scalar‘𝑆) & ⊢ 𝐾 = (Base‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → ℚ ⊆ 𝐾) & ⊢ · = ( ·𝑠 ‘𝑆) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ≠ (0g‘𝑆)) & ⊢ (𝜑 → ((𝑟 · 𝐵) ∈ 𝑉 → ((𝐿‘(𝑟 · 𝐵)) < 𝑅 → ((𝑀‘(𝐹‘(𝑟 · 𝐵))) / 𝑅) ≤ 𝐴))) & ⊢ (𝜑 → ¬ (𝑀‘(𝐹‘𝐵)) ≤ (𝐴 · (𝐿‘𝐵))) ⇒ ⊢ ¬ 𝜑 | ||
29-Sep-2021 | clmsub4 22714 | Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 5-Aug-2007.) (Revised by AV, 29-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 + 𝐵) + (-1 · (𝐶 + 𝐷))) = ((𝐴 + (-1 · 𝐶)) + (𝐵 + (-1 · 𝐷)))) | ||
29-Sep-2021 | clmnegsubdi2 22713 | Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 29-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴))) | ||
28-Sep-2021 | clmvslinv 22716 | Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ((-1 · 𝐴) + 𝐴) = 0 ) | ||
28-Sep-2021 | clmvsrinv 22715 | A vector minus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + (-1 · 𝐴)) = 0 ) | ||
28-Sep-2021 | clmvsdi 22700 | Distributive law for scalar product (left-distributivity). (lmodvsdi 18709 analog.) (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌))) | ||
28-Sep-2021 | clmvscl 22696 | Closure of scalar product for a complex module. (lmodvscl 18703 analog.) (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉) | ||
28-Sep-2021 | clmneg1 22690 | Minus one is in the scalar ring of a complex module. (Contributed by AV, 28-Sep-2021.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂMod → -1 ∈ 𝐾) | ||
24-Sep-2021 | imaex 6996 | The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by JJ, 24-Sep-2021.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 “ 𝐵) ∈ V | ||
24-Sep-2021 | ordelinel 5742 | The intersection of two ordinal classes is an element of a third if and only if either one of them is. (Contributed by David Moews, 1-May-2017.) (Proof shortened by JJ, 24-Sep-2021.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∩ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶))) | ||
24-Sep-2021 | ordnbtwn 5733 | There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 21-Jun-1998.) (Proof shortened by JJ, 24-Sep-2021.) |
⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)) | ||
24-Sep-2021 | suctr 5725 | The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) (Proof shortened by JJ, 24-Sep-2021.) |
⊢ (Tr 𝐴 → Tr suc 𝐴) | ||
24-Sep-2021 | ordtr3 5686 | Transitive law for ordinal classes. (Contributed by Mario Carneiro, 30-Dec-2014.) (Proof shortened by JJ, 24-Sep-2021.) |
⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) | ||
24-Sep-2021 | ordtri3 5676 | A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by JJ, 24-Sep-2021.) |
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴))) | ||
21-Sep-2021 | cncvs 22753 | The set of complex numbers (as a subring of itself) is a complex vector space. The vector operation is +, and the scalar product is ·. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 21-Sep-2021.) |
⊢ 𝐶 = (ringLMod‘ℂfld) ⇒ ⊢ 𝐶 ∈ ℂVec | ||
21-Sep-2021 | cnrlvec 22752 | The set of complex numbers (as a subring of itself) is a (left) vector space over itself. The vector operation is +, and the scalar product is ·. (Contributed by AV, 21-Sep-2021.) |
⊢ 𝐶 = (ringLMod‘ℂfld) ⇒ ⊢ 𝐶 ∈ LVec | ||
21-Sep-2021 | cnrlmod 22751 | The set of complex numbers (as a subring of itself) is a left module over itself. The vector operation is +, and the scalar product is ·. (Contributed by AV, 21-Sep-2021.) |
⊢ 𝐶 = (ringLMod‘ℂfld) ⇒ ⊢ 𝐶 ∈ LMod | ||
21-Sep-2021 | cnrbas 22750 | The set of complex numbers is the base set of the complex numbers as left module induced by the complex numbers as subring of itself. (Contributed by AV, 21-Sep-2021.) |
⊢ 𝐶 = (ringLMod‘ℂfld) ⇒ ⊢ (Base‘𝐶) = ℂ | ||
21-Sep-2021 | cvsi 22738 | The properties of a complex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. (Contributed by NM, 3-Nov-2006.) (Revised by AV, 21-Sep-2021.) |
⊢ 𝑋 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑆 = (Base‘(Scalar‘𝑊)) & ⊢ ∙ = ( ·sf ‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ (𝑊 ∈ ℂVec → (𝑊 ∈ Abel ∧ (𝑆 ⊆ ℂ ∧ ∙ :(𝑆 × 𝑋)⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑆 (∀𝑧 ∈ 𝑋 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝑆 (((𝑦 + 𝑧) · 𝑥) = ((𝑦 · 𝑥) + (𝑧 · 𝑥)) ∧ ((𝑦 · 𝑧) · 𝑥) = (𝑦 · (𝑧 · 𝑥))))))) | ||
21-Sep-2021 | clmnegneg 22712 | Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 21-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (-1 · (-1 · 𝐴)) = 𝐴) | ||
21-Sep-2021 | clmpm1dir 22711 | Subtractive distributive law for the scalar product of a complex left module. (Contributed by NM, 31-Jul-2007.) (Revised by AV, 21-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐾 = (Base‘(Scalar‘𝑊)) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (-1 · (𝐵 · 𝐶)))) | ||
21-Sep-2021 | clmvs2 22702 | A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (Revised by AV, 21-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + 𝐴) = (2 · 𝐴)) | ||
20-Sep-2021 | cnstrcvs 22749 | The set of complex numbers is a complex vector space. The vector operation is +, and the scalar product is ·. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 20-Sep-2021.) |
⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) ⇒ ⊢ 𝑊 ∈ ℂVec | ||
20-Sep-2021 | cnlmod 22748 | The set of complex numbers is a left module over itself. The vector operation is +, and the scalar product is ·. (Contributed by AV, 20-Sep-2021.) |
⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) ⇒ ⊢ 𝑊 ∈ LMod | ||
20-Sep-2021 | cnlmod4 22747 | Lemma 4 for cnlmod 22748. (Contributed by AV, 20-Sep-2021.) |
⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) ⇒ ⊢ ( ·𝑠 ‘𝑊) = · | ||
20-Sep-2021 | cnlmodlem3 22746 | Lemma 3 for cnlmod 22748. (Contributed by AV, 20-Sep-2021.) |
⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) ⇒ ⊢ (Scalar‘𝑊) = ℂfld | ||
20-Sep-2021 | cnlmodlem2 22745 | Lemma 2 for cnlmod 22748. (Contributed by AV, 20-Sep-2021.) |
⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) ⇒ ⊢ (+g‘𝑊) = + | ||
20-Sep-2021 | cnlmodlem1 22744 | Lemma 1 for cnlmod 22748. (Contributed by AV, 20-Sep-2021.) |
⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) ⇒ ⊢ (Base‘𝑊) = ℂ | ||
19-Sep-2021 | mpfsubrg 19353 | Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) (Revised by AV, 19-Sep-2021.) |
⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (SubRing‘(𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))) | ||
18-Sep-2021 | opidg 40316 | The ordered pair 〈𝐴, 𝐴〉 in Kuratowski's representation. Closed form of opid 4359. (Contributed by AV, 18-Sep-2020.) (Revised by AV, 18-Sep-2021.) |
⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 = {{𝐴}}) | ||
18-Sep-2021 | evlsvar 19344 | Polynomial evaluation maps variables to projections. (Contributed by Stefan O'Rear, 12-Mar-2015.) (Proof shortened by AV, 18-Sep-2021.) |
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ 𝑉 = (𝐼 mVar 𝑈) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑋))) | ||
18-Sep-2021 | evlssca 19343 | Polynomial evaluation maps scalars to constant functions. (Contributed by Stefan O'Rear, 13-Mar-2015.) (Proof shortened by AV, 18-Sep-2021.) |
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ 𝑊 = (𝐼 mPoly 𝑈) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑋 ∈ 𝑅) ⇒ ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑𝑚 𝐼) × {𝑋})) | ||
18-Sep-2021 | evlsval2 19341 | Characterizing properties of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 12-Mar-2015.) (Revised by AV, 18-Sep-2021.) |
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ 𝑊 = (𝐼 mPoly 𝑈) & ⊢ 𝑉 = (𝐼 mVar 𝑈) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑𝑚 𝐼)) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝑋 = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥})) & ⊢ 𝑌 = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥))) ⇒ ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄 ∘ 𝐴) = 𝑋 ∧ (𝑄 ∘ 𝑉) = 𝑌))) | ||
18-Sep-2021 | evlsval 19340 | Value of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 11-Mar-2015.) (Revised by AV, 18-Sep-2021.) |
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ 𝑊 = (𝐼 mPoly 𝑈) & ⊢ 𝑉 = (𝐼 mVar 𝑈) & ⊢ 𝑈 = (𝑆 ↾s 𝑅) & ⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑𝑚 𝐼)) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝑋 = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥})) & ⊢ 𝑌 = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥))) ⇒ ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (℩𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = 𝑌))) | ||
18-Sep-2021 | hashtpg 13121 | The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 18-Sep-2021.) |
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) ↔ (#‘{𝐴, 𝐵, 𝐶}) = 3)) | ||
18-Sep-2021 | hashprg 13043 | The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ 𝐵 ↔ (#‘{𝐴, 𝐵}) = 2)) | ||
18-Sep-2021 | pwfseqlem2 9360 | Lemma for pwfseq 9365. (Contributed by Mario Carneiro, 18-Nov-2014.) (Revised by AV, 18-Sep-2021.) |
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛)) & ⊢ (𝜑 → 𝑋 ⊆ 𝐴) & ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋) & ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑𝑚 𝑛)–1-1→𝑥) & ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}) & ⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))) ⇒ ⊢ ((𝑌 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌))) | ||
18-Sep-2021 | hsmexlem2 9132 | Lemma for hsmex 9137. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9276 but use of order types allows to canonically choose the sub-bijections, removing the choice requirement. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by AV, 18-Sep-2021.) |
⊢ 𝐹 = OrdIso( E , 𝐵) & ⊢ 𝐺 = OrdIso( E , ∪ 𝑎 ∈ 𝐴 𝐵) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶))) | ||
18-Sep-2021 | fopwdom 7953 | Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) (Revised by AV, 18-Sep-2021.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴) | ||
18-Sep-2021 | preqr1g 4325 | Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 4319. (Contributed by AV, 29-Jan-2021.) (Revised by AV, 18-Sep-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)) | ||
18-Sep-2021 | nfbidf 2079 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒)) | ||
18-Sep-2021 | 19.21 2062 | Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." See 19.21v 1855 for a version requiring fewer axioms. See also 19.21h 2107. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) | ||
18-Sep-2021 | nfan1 2056 | A closed form of nfan 1816. (Contributed by Mario Carneiro, 3-Oct-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) | ||
18-Sep-2021 | nfim1 2055 | A closed form of nfim 1813. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ Ⅎ𝑥(𝜑 → 𝜓) | ||
18-Sep-2021 | nf5dv 2012 | Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
18-Sep-2021 | nfimd 1812 | If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓 → 𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒)) | ||
18-Sep-2021 | nfn 1768 | Inference associated with nfnt 1767. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥 ¬ 𝜑 | ||
17-Sep-2021 | esum2dlem 29481 | Lemma for esum2d 29482 (finite case). (Contributed by Thierry Arnoux, 17-May-2020.) (Proof shortened by AV, 17-Sep-2021.) |
⊢ Ⅎ𝑘𝐹 & ⊢ (𝑧 = 〈𝑗, 𝑘〉 → 𝐹 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹) | ||
17-Sep-2021 | 2spotmdisj 26595 | The sets of paths of length 2 with a given vertex in the middle are distinct for different vertices in the middle. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 17-Sep-2021.) |
⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑎)}) ⇒ ⊢ (𝑉 ∈ 𝑊 → Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥)) | ||
17-Sep-2021 | ustssel 21819 | A uniform structure is upward closed. Condition FI of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.) (Proof shortened by AV, 17-Sep-2021.) |
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑊 ⊆ (𝑋 × 𝑋)) → (𝑉 ⊆ 𝑊 → 𝑊 ∈ 𝑈)) | ||
17-Sep-2021 | isust 21817 | The predicate "𝑈 is a uniform structure with base 𝑋." (Contributed by Thierry Arnoux, 15-Nov-2017.) (Revised by AV, 17-Sep-2021.) |
⊢ (𝑋 ∈ 𝑉 → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣))))) | ||
17-Sep-2021 | ustval 21816 | The class of all uniform structures for a base 𝑋. (Contributed by Thierry Arnoux, 15-Nov-2017.) (Revised by AV, 17-Sep-2021.) |
⊢ (𝑋 ∈ 𝑉 → (UnifOn‘𝑋) = {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))}) | ||
17-Sep-2021 | nfim 1813 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1701 changed. (Revised by Wolf Lammen, 17-Sep-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 → 𝜓) | ||
16-Sep-2021 | 127prm 40053 | 127 is a prime number. (Contributed by AV, 16-Aug-2021.) (Proof shortened by AV, 16-Sep-2021.) |
⊢ ;;127 ∈ ℙ | ||
16-Sep-2021 | log2ub 24476 | log2 is less than 253 / 365. If written in decimal, this is because log2 = 0.693147... is less than 253/365 = 0.693151... , so this is a very tight bound, at five decimal places. (Contributed by Mario Carneiro, 7-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
⊢ (log‘2) < (;;253 / ;;365) | ||
16-Sep-2021 | 4001prm 15690 | 4001 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
⊢ 𝑁 = ;;;4001 ⇒ ⊢ 𝑁 ∈ ℙ | ||
16-Sep-2021 | 4001lem4 15689 | Lemma for 4001prm 15690. Calculate the GCD of 2↑800 − 1≡2310 with 𝑁 = 4001. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
⊢ 𝑁 = ;;;4001 ⇒ ⊢ (((2↑;;800) − 1) gcd 𝑁) = 1 | ||
16-Sep-2021 | 4001lem3 15688 | Lemma for 4001prm 15690. Calculate a power mod. In decimal, we calculate 2↑1000 = 2↑800 · 2↑200≡2311 · 902 = 521𝑁 + 1 and finally 2↑(𝑁 − 1) = (2↑1000)↑4≡1↑4 = 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
⊢ 𝑁 = ;;;4001 ⇒ ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁) | ||
16-Sep-2021 | 4001lem2 15687 | Lemma for 4001prm 15690. Calculate a power mod. In decimal, we calculate 2↑400 = (2↑200)↑2≡902↑2 = 203𝑁 + 1401 and 2↑800 = (2↑400)↑2≡1401↑2 = 490𝑁 + 2311 ≡2311. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
⊢ 𝑁 = ;;;4001 ⇒ ⊢ ((2↑;;800) mod 𝑁) = (;;;2311 mod 𝑁) | ||
16-Sep-2021 | 4001lem1 15686 | Lemma for 4001prm 15690. Calculate a power mod. In decimal, we calculate 2↑12 = 4096 = 𝑁 + 95, 2↑24 = (2↑12)↑2≡95↑2 = 2𝑁 + 1023, 2↑25 = 2↑24 · 2≡1023 · 2 = 2046, 2↑50 = (2↑25)↑2≡2046↑2 = 1046𝑁 + 1070, 2↑100 = (2↑50)↑2≡1070↑2 = 286𝑁 + 614 and 2↑200 = (2↑100)↑2≡614↑2 = 94𝑁 + 902 ≡902. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
⊢ 𝑁 = ;;;4001 ⇒ ⊢ ((2↑;;200) mod 𝑁) = (;;902 mod 𝑁) | ||
16-Sep-2021 | 2503lem2 15683 | Lemma for 2503prm 15685. Calculate a power mod. We calculate 2↑19 = 2↑18 · 2≡1832 · 2 = 𝑁 + 1161, 2↑38 = (2↑19)↑2≡1161↑2 = 538𝑁 + 1307, 2↑39 = 2↑38 · 2≡1307 · 2 = 𝑁 + 111, 2↑78 = (2↑39)↑2≡111↑2 = 5𝑁 − 194, 2↑156 = (2↑78)↑2≡194↑2 = 15𝑁 + 91, 2↑312 = (2↑156)↑2≡91↑2 = 3𝑁 + 772, 2↑624 = (2↑312)↑2≡772↑2 = 238𝑁 + 270, 2↑1248 = (2↑624)↑2≡270↑2 = 29𝑁 + 313, 2↑1251 = 2↑1248 · 8≡313 · 8 = 𝑁 + 1 and finally 2↑(𝑁 − 1) = (2↑1251)↑2≡1↑2 = 1. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
⊢ 𝑁 = ;;;2503 ⇒ ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁) | ||
16-Sep-2021 | 2503lem1 15682 | Lemma for 2503prm 15685. Calculate a power mod. In decimal, we calculate 2↑18 = 512↑2 = 104𝑁 + 1832≡1832. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
⊢ 𝑁 = ;;;2503 ⇒ ⊢ ((2↑;18) mod 𝑁) = (;;;1832 mod 𝑁) | ||
16-Sep-2021 | 1259lem4 15679 | Lemma for 1259prm 15681. Calculate a power mod. In decimal, we calculate 2↑306 = (2↑76)↑4 · 4≡5↑4 · 4 = 2𝑁 − 18, 2↑612 = (2↑306)↑2≡18↑2 = 324, 2↑629 = 2↑612 · 2↑17≡324 · 136 = 35𝑁 − 1 and finally 2↑(𝑁 − 1) = (2↑629)↑2≡1↑2 = 1. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
⊢ 𝑁 = ;;;1259 ⇒ ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁) | ||
16-Sep-2021 | 1259lem3 15678 | Lemma for 1259prm 15681. Calculate a power mod. In decimal, we calculate 2↑38 = 2↑34 · 2↑4≡870 · 16 = 11𝑁 + 71 and 2↑76 = (2↑34)↑2≡71↑2 = 4𝑁 + 5≡5. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
⊢ 𝑁 = ;;;1259 ⇒ ⊢ ((2↑;76) mod 𝑁) = (5 mod 𝑁) | ||
16-Sep-2021 | 1259lem1 15676 | Lemma for 1259prm 15681. Calculate a power mod. In decimal, we calculate 2↑16 = 52𝑁 + 68≡68 and 2↑17≡68 · 2 = 136 in this lemma. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.) |
⊢ 𝑁 = ;;;1259 ⇒ ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁) | ||
16-Sep-2021 | decrmac 11453 | Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by AV, 16-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐺 ∈ ℕ0 & ⊢ ((𝐴 · 𝑃) + 𝐺) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝑁) = ;𝐺𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 | ||
16-Sep-2021 | decrmanc 11452 | Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by AV, 16-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑃 ∈ ℕ0 & ⊢ (𝐴 · 𝑃) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝑁) = 𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 | ||
16-Sep-2021 | nf5-1 2010 | One direction of nf5 2102 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑) | ||
16-Sep-2021 | nfntht2 1711 | Closed form of nfnth 1719. (Contributed by BJ, 16-Sep-2021.) |
⊢ (∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑) | ||
16-Sep-2021 | nfntht 1710 | Closed form of nfnth 1719. (Contributed by BJ, 16-Sep-2021.) |
⊢ (¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑) | ||
16-Sep-2021 | nftht0 1709 | Closed form of nfth 1718. (Contributed by Wolf Lammen, 19-Aug-2018.) (Proof shortened by BJ, 16-Sep-2021.) |
⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑) | ||
16-Sep-2021 | nfd 1707 | Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Wolf Lammen, 16-Sep-2021.) |
⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
16-Sep-2021 | nfri 1706 | Consequence of the definition of not-free. (Contributed by Wolf Lammen, 16-Sep-2021.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑥𝜑 → ∀𝑥𝜑) | ||
16-Sep-2021 | nf4 1704 | Alternate definition of non-freeness. This definition uses only primitive symbols. (Contributed by BJ, 16-Sep-2021.) |
⊢ (Ⅎ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑)) | ||
16-Sep-2021 | nf3 1703 | Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021.) |
⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) | ||
16-Sep-2021 | nf2 1702 | Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021.) |
⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑)) | ||
15-Sep-2021 | evengpoap3 40215 | If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of an odd Goldbach number and 3. (Contributed by AV, 27-Jul-2020.) (Proof shortened by AV, 15-Sep-2021.) |
⊢ (∀𝑚 ∈ Odd (7 < 𝑚 → 𝑚 ∈ GoldbachOddALTV ) → ((𝑁 ∈ (ℤ≥‘;12) ∧ 𝑁 ∈ Even ) → ∃𝑜 ∈ GoldbachOddALTV 𝑁 = (𝑜 + 3))) | ||
15-Sep-2021 | bposlem9 24817 | Lemma for bpos 24818. Derive a contradiction. (Contributed by Mario Carneiro, 14-Mar-2014.) (Proof shortened by AV, 15-Sep-2021.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((((√‘2) · (𝐺‘(√‘𝑛))) + ((9 / 4) · (𝐺‘(𝑛 / 2)))) + ((log‘2) / (√‘(2 · 𝑛))))) & ⊢ 𝐺 = (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) / 𝑥)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → ;64 < 𝑁) & ⊢ (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) ⇒ ⊢ (𝜑 → 𝜓) | ||
15-Sep-2021 | bposlem5 24813 | Lemma for bpos 24818. Bound the product of all small primes in the binomial coefficient. (Contributed by Mario Carneiro, 15-Mar-2014.) (Proof shortened by AV, 15-Sep-2021.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘5)) & ⊢ (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt ((2 · 𝑁)C𝑁))), 1)) & ⊢ 𝐾 = (⌊‘((2 · 𝑁) / 3)) & ⊢ 𝑀 = (⌊‘(√‘(2 · 𝑁))) ⇒ ⊢ (𝜑 → (seq1( · , 𝐹)‘𝑀) ≤ ((2 · 𝑁)↑𝑐(((√‘(2 · 𝑁)) / 3) + 2))) | ||
15-Sep-2021 | bpos1 24808 | Bertrand's postulate, checked numerically for 𝑁 ≤ 64, using the prime sequence 2, 3, 5, 7, 13, 23, 43, 83. (Contributed by Mario Carneiro, 12-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ≤ ;64) → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) | ||
15-Sep-2021 | log2ublem3 24475 | Lemma for log2ub 24476. In decimal, this is a proof that the first four terms of the series for log2 is less than 53056 / 76545. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.) |
⊢ (((3↑7) · (5 · 7)) · Σ𝑛 ∈ (0...3)(2 / ((3 · ((2 · 𝑛) + 1)) · (9↑𝑛)))) ≤ ;;;;53056 | ||
15-Sep-2021 | odrngstr 15889 | Functionality of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.) (Proof shortened by AV, 15-Sep-2021.) |
⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(TopSet‘ndx), 𝐽〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), 𝐷〉}) ⇒ ⊢ 𝑊 Struct 〈1, ;12〉 | ||
15-Sep-2021 | 2503lem3 15684 | Lemma for 2503prm 15685. Calculate the GCD of 2↑18 − 1≡1831 with 𝑁 = 2503. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.) |
⊢ 𝑁 = ;;;2503 ⇒ ⊢ (((2↑;18) − 1) gcd 𝑁) = 1 | ||
15-Sep-2021 | 1259lem2 15677 | Lemma for 1259prm 15681. Calculate a power mod. In decimal, we calculate 2↑34 = (2↑17)↑2≡136↑2≡14𝑁 + 870. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.) |
⊢ 𝑁 = ;;;1259 ⇒ ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁) | ||
15-Sep-2021 | hbe1a 2009 | Dual statement of hbe1 2008. Modified version of axc7e 2118 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.) |
⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) | ||
15-Sep-2021 | nfi 1705 | Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Wolf Lammen, 15-Sep-2021.) |
⊢ (∃𝑥𝜑 → ∀𝑥𝜑) ⇒ ⊢ Ⅎ𝑥𝜑 | ||
14-Sep-2021 | wl-nf2-nf 32464 | hba1 2137 is sufficient to let a df-nf 1701 style definition be stricter than nf5 2102. (Contributed by Wolf Lammen, 14-Sep-2021.) |
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) ⇒ ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑)) | ||
14-Sep-2021 | wl-nf-nf2 32463 | By ax-10 2006 the definition nf5 2102 is stricter than the df-nf 1701 style. (Contributed by Wolf Lammen, 14-Sep-2021.) |
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) ⇒ ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑)) | ||
12-Sep-2021 | nfv 1830 | If 𝑥 is not present in 𝜑, then 𝑥 is not free in 𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Definition change. (Revised by Wolf Lammen, 12-Sep-2021.) |
⊢ Ⅎ𝑥𝜑 | ||
12-Sep-2021 | nfbii 1770 | Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 12-Sep-2021.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓) | ||
12-Sep-2021 | nfnth 1719 | No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 12-Sep-2021.) |
⊢ ¬ 𝜑 ⇒ ⊢ Ⅎ𝑥𝜑 | ||
12-Sep-2021 | nfth 1718 | No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 12-Sep-2021.) |
⊢ 𝜑 ⇒ ⊢ Ⅎ𝑥𝜑 | ||
11-Sep-2021 | wl-section-nf 32462 |
The current definition of 'not free', nf5 2102, has its downsides. In
particular, it often drags axioms ax-10 2006 and ax-12 2034 into proofs, that
are not needed otherwise. This is because of the particular structure
of the term ∀𝑥(𝜑 → ∀𝑥𝜑). It does not allow an easy
transition 𝜑--> ¬ 𝜑 (see nfn 1768).
The mix of both
quantified and simple 𝜑 requires explicit use of sp 2041 (or
ax-12 2034)
in many instances. And, finally, the nesting of the quantifier ∀𝑥
sometimes requires an invocation of theorems like nfa1 2015.
All of this
mandates the use of ax-10 2006 and/or ax-12 2034.
On the other hand, the current definition is structurally better aligned with both the hb* series of theorems and ax-5 1827. A note on ax-10 2006: The obvious content of this axiom is, that the ¬ operator does not change the not-free state of a set varable. This is in fact only one aspect of this axiom. The other one, more hidden, states that 𝑥 is not free in ∀𝑥𝜑. A simple transformation renders this axiom as (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑), from which the second aspect is better deduced. Both aspects are not needed in simple applications of the df-nf 1701 style definition, while use of the current nf5 2102 incurs these. A note on ax-12 2034: This axiom enters proofs using Ⅎ via sp 2041 or 19.8a 2039 or their variants. In a context where both mixed quantified and simple variables 𝜑 appear (like 19.21 2062), this axiom is almost always required, no matter how 'not free' is defined. But in a context, where a variable 𝜑 appears only quantified, chances are, this axiom can be evaded when using df-nf 1701, but not when using nf5 2102. (Contributed by Wolf Lammen, 11-Sep-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝜑 ⇒ ⊢ 𝜑 | ||
11-Sep-2021 | nfa1OLD 2143 | Obsolete proof of nfa1 2015 as of 12-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥∀𝑥𝜑 | ||
11-Sep-2021 | nf5 2102 | Alternate definition of df-nf 1701. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | ||
11-Sep-2021 | 19.23t 2066 | Closed form of Theorem 1977.23 of [Margaris] p. 90. See 19.23 2067. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) | ||
11-Sep-2021 | 19.21t 2061 | Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2062. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) | ||
11-Sep-2021 | 19.9d 2058 | A deduction version of one direction of 19.9 2060. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
⊢ (𝜓 → Ⅎ𝑥𝜑) ⇒ ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑)) | ||
11-Sep-2021 | nf5r 2052 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | ||
10-Sep-2021 | deccarry 39941 | Add 1 to a 2 digit number with carry. This is a special case of decsucc 11426, but in closed form. As observed by ML, this theorem allows for carrying the 1 down multiple decimal constructors, so we can carry the 1 multiple times down a multi-digit number, e.g. by applying this theorem three times we get (;;999 + 1) = ;;;1000. (Contributed by AV, 4-Aug-2020.) (Revised by ML, 8-Aug-2020.) (Proof shortened by AV, 10-Sep-2021.) |
⊢ (𝐴 ∈ ℕ → (;𝐴9 + 1) = ;(𝐴 + 1)0) | ||
9-Sep-2021 | dpfrac1 42312 | Prove a simple equivalence involving the decimal point. See df-dp 42308 and dpcl 42311. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = (;𝐴𝐵 / ;10)) | ||
9-Sep-2021 | df-dp2 42306 | Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 11370. (Contributed by David A. Wheeler, 15-May-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | ||
9-Sep-2021 | tgoldbach 40232 | The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 40230 and ax-tgoldbachgt 40231. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOddALTV ) | ||
9-Sep-2021 | ax-tgoldbachgt 40231 | The ternary Goldbach conjecture is valid for big odd numbers (i.e. for all odd numbers greater than a fixed 𝑚). This is proven by Helfgott (see section 7.4 in [Helfgott] p. 70) for m = 10^27. Temporarily provided as "axiom". (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ Odd (𝑚 < 𝑛 → 𝑛 ∈ GoldbachOddALTV )) | ||
9-Sep-2021 | tgoldbachlt 40230 | The ternary Goldbach conjecture is valid for small odd numbers (i.e. for all odd numbers less than a fixed big 𝑚 greater than 8 x 10^30). This is verified for m = 8.875694 x 10^30 by Helfgott, see tgblthelfgott 40229. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ∃𝑚 ∈ ℕ ((8 · (;10↑;30)) < 𝑚 ∧ ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOddALTV )) | ||
9-Sep-2021 | tgblthelfgott 40229 | The ternary Goldbach conjecture is valid for all odd numbers less than 8.8 x 10^30 (actually 8.875694 x 10^30, see section 1.2.2 in [Helfgott] p. 4, using bgoldbachlt 40227, ax-hgprmladder 40228 and bgoldbtbnd 40225. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (;88 · (;10↑;29))) → 𝑁 ∈ GoldbachOddALTV ) | ||
9-Sep-2021 | ax-hgprmladder 40228 | There is a partition ("ladder") of primes from 7 to 8.8 x 10^30 with parts ("rungs") having lengths of at least 4 and at most N - 4, see section 1.2.2 in [Helfgott] p. 4. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ∃𝑑 ∈ (ℤ≥‘3)∃𝑓 ∈ (RePart‘𝑑)(((𝑓‘0) = 7 ∧ (𝑓‘1) = ;13 ∧ (𝑓‘𝑑) = (;89 · (;10↑;29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓‘𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖)) < ((4 · (;10↑;18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖)))) | ||
9-Sep-2021 | bgoldbachlt 40227 | The binary Goldbach conjecture is valid for small even numbers (i.e. for all even numbers less than or equal to a fixed big 𝑚). This is verified for m = 4 x 10^18 by Oliveira e Silva, see ax-bgbltosilva 40226. (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ∃𝑚 ∈ ℕ ((4 · (;10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )) | ||
9-Sep-2021 | ax-bgbltosilva 40226 | The binary Goldbach conjecture is valid for all even numbers less than or equal to 4x10^18, see result of [OeSilva] p. ?. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 ≤ (4 · (;10↑;18))) → 𝑁 ∈ GoldbachEven ) | ||
9-Sep-2021 | 1t10e1p1e11 39937 | 11 is 1 times 10 to the power of 1, plus 1. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ;11 = ((1 · (;10↑1)) + 1) | ||
9-Sep-2021 | problem2 30813 | Practice problem 2. Clues: oveq12i 6561 adddiri 9930 add4i 10139 mulcli 9924 recni 9931 2re 10967 3eqtri 2636 10re 11393 5re 10976 1re 9918 4re 10974 eqcomi 2619 5p4e9 11044 oveq1i 6559 df-3 10957. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Revised by AV, 9-Sep-2021.) (Proof modification is discouraged.) |
⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = ((3 · ;10) + 9) | ||
9-Sep-2021 | znbaslem 19705 | Lemma for znbas 19711. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 9-Sep-2021.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐸 = Slot 𝐾 & ⊢ 𝐾 ∈ ℕ & ⊢ 𝐾 < ;10 ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌)) | ||
9-Sep-2021 | opsrbaslem 19298 | Get a component of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) & ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝑁 < ;10 ⇒ ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂)) | ||
9-Sep-2021 | otpsle 15877 | The order of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ 𝐾 = {〈(Base‘ndx), 𝐵〉, 〈(TopSet‘ndx), 𝐽〉, 〈(le‘ndx), ≤ 〉} ⇒ ⊢ ( ≤ ∈ 𝑉 → ≤ = (le‘𝐾)) | ||
9-Sep-2021 | otpstset 15876 | The open sets of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ 𝐾 = {〈(Base‘ndx), 𝐵〉, 〈(TopSet‘ndx), 𝐽〉, 〈(le‘ndx), ≤ 〉} ⇒ ⊢ (𝐽 ∈ 𝑉 → 𝐽 = (TopSet‘𝐾)) | ||
9-Sep-2021 | otpsbas 15875 | The base set of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ 𝐾 = {〈(Base‘ndx), 𝐵〉, 〈(TopSet‘ndx), 𝐽〉, 〈(le‘ndx), ≤ 〉} ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐾)) | ||
9-Sep-2021 | otpsstr 15874 | Functionality of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ 𝐾 = {〈(Base‘ndx), 𝐵〉, 〈(TopSet‘ndx), 𝐽〉, 〈(le‘ndx), ≤ 〉} ⇒ ⊢ 𝐾 Struct 〈1, ;10〉 | ||
9-Sep-2021 | pleid 15872 | Utility theorem: self-referencing, index-independent form of df-ple 15788. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.) |
⊢ le = Slot (le‘ndx) | ||
9-Sep-2021 | plendx 15870 | Index value of the df-ple 15788 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ (le‘ndx) = ;10 | ||
9-Sep-2021 | df-ple 15788 | Define less-than-or-equal ordering extractor for posets and related structures. We use ;10 for the index to avoid conflict with 1 through 9 used for other purposes. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ le = Slot ;10 | ||
9-Sep-2021 | karatsuba 15630 | The Karatsuba multiplication algorithm. If 𝑋 and 𝑌 are decomposed into two groups of digits of length 𝑀 (only the lower group is known to be this size but the algorithm is most efficient when the partition is chosen near the middle of the digit string), then 𝑋𝑌 can be written in three groups of digits, where each group needs only one multiplication. Thus, we can halve both inputs with only three multiplications on the smaller operands, yielding an asymptotic improvement of n^(log2 3) instead of n^2 for the "naive" algorithm decmul1c 11463. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑆 ∈ ℕ0 & ⊢ 𝑀 ∈ ℕ0 & ⊢ (𝐴 · 𝐶) = 𝑅 & ⊢ (𝐵 · 𝐷) = 𝑇 & ⊢ ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = ((𝑅 + 𝑆) + 𝑇) & ⊢ ((𝐴 · (;10↑𝑀)) + 𝐵) = 𝑋 & ⊢ ((𝐶 · (;10↑𝑀)) + 𝐷) = 𝑌 & ⊢ ((𝑅 · (;10↑𝑀)) + 𝑆) = 𝑊 & ⊢ ((𝑊 · (;10↑𝑀)) + 𝑇) = 𝑍 ⇒ ⊢ (𝑋 · 𝑌) = 𝑍 | ||
8-Sep-2021 | decsplit 15625 | Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑀 ∈ ℕ0 & ⊢ (𝑀 + 1) = 𝑁 & ⊢ ((𝐴 · (;10↑𝑀)) + 𝐵) = 𝐶 ⇒ ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ;𝐶𝐷 | ||
8-Sep-2021 | decsplit1 15624 | Split a decimal number into two parts. Base case: 𝑁 = 1. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ ((𝐴 · (;10↑1)) + 𝐵) = ;𝐴𝐵 | ||
8-Sep-2021 | decsplit0 15623 | Split a decimal number into two parts. Base case: 𝑁 = 0. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ ((𝐴 · (;10↑0)) + 0) = 𝐴 | ||
8-Sep-2021 | decsplit0b 15622 | Split a decimal number into two parts. Base case: 𝑁 = 0. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ ((𝐴 · (;10↑0)) + 𝐵) = (𝐴 + 𝐵) | ||
8-Sep-2021 | 3dvdsdec 14892 | A decimal number is divisible by three iff the sum of its two "digits" is divisible by three. The term "digits" in its narrow sense is only correct if 𝐴 and 𝐵 actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers 𝐴 and 𝐵, especially if 𝐴 is itself a decimal number, e.g. 𝐴 = ;𝐶𝐷. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 ⇒ ⊢ (3 ∥ ;𝐴𝐵 ↔ 3 ∥ (𝐴 + 𝐵)) | ||
8-Sep-2021 | 3dvds 14890 | A rule for divisibility by 3 of a number written in base 10. This is Metamath 100 proof #85. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(0...𝑁)⟶ℤ) → (3 ∥ Σ𝑘 ∈ (0...𝑁)((𝐹‘𝑘) · (;10↑𝑘)) ↔ 3 ∥ Σ𝑘 ∈ (0...𝑁)(𝐹‘𝑘))) | ||
8-Sep-2021 | 0.999... 14451 | The recurring decimal 0.999..., which is defined as the infinite sum 0.9 + 0.09 + 0.009 + ... i.e. 9 / 10↑1 + 9 / 10↑2 + 9 / 10↑3 + ..., is exactly equal to 1, according to ZF set theory. Interestingly, about 40% of the people responding to a poll at http://forum.physorg.com/index.php?showtopic=13177 disagree. (Contributed by NM, 2-Nov-2007.) (Revised by AV, 8-Sep-2021.) |
⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 | ||
8-Sep-2021 | 1lt10 11557 | 1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 1 < ;10 | ||
8-Sep-2021 | 2lt10 11556 | 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 2 < ;10 | ||
8-Sep-2021 | 3lt10 11555 | 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 3 < ;10 | ||
8-Sep-2021 | 4lt10 11554 | 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 4 < ;10 | ||
8-Sep-2021 | 5lt10 11553 | 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 5 < ;10 | ||
8-Sep-2021 | 6lt10 11552 | 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 6 < ;10 | ||
8-Sep-2021 | 7lt10 11551 | 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 7 < ;10 | ||
8-Sep-2021 | 8lt10 11550 | 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 8 < ;10 | ||
8-Sep-2021 | 9lt10 11549 | 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.) |
⊢ 9 < ;10 | ||
8-Sep-2021 | decltdi 11423 | Comparing a digit to a decimal integer. (Contributed by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐶 ≤ 9 ⇒ ⊢ 𝐶 < ;𝐴𝐵 | ||
8-Sep-2021 | decleh 11417 | Comparing two decimal integers (unequal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐶 ≤ 9 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ;𝐴𝐶 ≤ ;𝐵𝐷 | ||
8-Sep-2021 | decle 11416 | Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐵 ≤ 𝐶 ⇒ ⊢ ;𝐴𝐵 ≤ ;𝐴𝐶 | ||
8-Sep-2021 | 3declth 11413 | Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐴 < 𝐵 & ⊢ 𝐶 ≤ 9 & ⊢ 𝐸 ≤ 9 ⇒ ⊢ ;;𝐴𝐶𝐸 < ;;𝐵𝐷𝐹 | ||
8-Sep-2021 | declth 11410 | Comparing two decimal integers (unequal higher places). (Contributed by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐶 ≤ 9 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ;𝐴𝐶 < ;𝐵𝐷 | ||
8-Sep-2021 | le9lt10 11405 | A "decimal digit" (i.e. a nonnegative integer less than or equal to 9) is less then 10. (Contributed by AV, 8-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐴 ≤ 9 ⇒ ⊢ 𝐴 < ;10 | ||
8-Sep-2021 | 10re 11393 | The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.) |
⊢ ;10 ∈ ℝ | ||
8-Sep-2021 | 10pos 11391 | The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 8-Sep-2021.) |
⊢ 0 < ;10 | ||
6-Sep-2021 | ex-prmo 26708 | Example for df-prmo 15574: (#p‘10) = 2 · 3 · 5 · 7. (Contributed by AV, 6-Sep-2021.) |
⊢ (#p‘;10) = ;;210 | ||
6-Sep-2021 | 10nprm 15658 | 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ ¬ ;10 ∈ ℙ | ||
6-Sep-2021 | 9t11e99 11547 | 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.) |
⊢ (9 · ;11) = ;99 | ||
6-Sep-2021 | 8t6e48 11535 | 8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 · 6) = ;48 | ||
6-Sep-2021 | 8t5e40 11533 | 8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 · 5) = ;40 | ||
6-Sep-2021 | 6t6e36 11522 | 6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (6 · 6) = ;36 | ||
6-Sep-2021 | 6t5e30 11520 | 6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (6 · 5) = ;30 | ||
6-Sep-2021 | 5t5e25 11515 | 5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (5 · 5) = ;25 | ||
6-Sep-2021 | 5t4e20 11513 | 5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (5 · 4) = ;20 | ||
6-Sep-2021 | 5t3e15 11511 | 5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (5 · 3) = ;15 | ||
6-Sep-2021 | 10m1e9 11506 | 10 - 1 = 9. (Contributed by AV, 6-Sep-2021.) |
⊢ (;10 − 1) = 9 | ||
6-Sep-2021 | 10p10e20 11504 | 10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (;10 + ;10) = ;20 | ||
6-Sep-2021 | 9p2e11 11495 | 9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (9 + 2) = ;11 | ||
6-Sep-2021 | 8p3e11 11488 | 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 + 3) = ;11 | ||
6-Sep-2021 | 8p2e10 11486 | 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.) |
⊢ (8 + 2) = ;10 | ||
6-Sep-2021 | 7p4e11 11481 | 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (7 + 4) = ;11 | ||
6-Sep-2021 | 7p3e10 11479 | 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.) |
⊢ (7 + 3) = ;10 | ||
6-Sep-2021 | 6p5e11 11476 | 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (6 + 5) = ;11 | ||
6-Sep-2021 | 6p4e10 11474 | 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.) |
⊢ (6 + 4) = ;10 | ||
6-Sep-2021 | 5p5e10 11472 | 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 6-Sep-2021.) |
⊢ (5 + 5) = ;10 | ||
6-Sep-2021 | decmul10add 11469 | A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑀 ∈ ℕ0 & ⊢ 𝐸 = (𝑀 · 𝐴) & ⊢ 𝐹 = (𝑀 · 𝐵) ⇒ ⊢ (𝑀 · ;𝐴𝐵) = (;𝐸0 + 𝐹) | ||
6-Sep-2021 | decmul2c 11465 | The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 = ;𝐴𝐵 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 & ⊢ (𝑃 · 𝐵) = ;𝐸𝐷 ⇒ ⊢ (𝑃 · 𝑁) = ;𝐶𝐷 | ||
6-Sep-2021 | decmul1c 11463 | The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 = ;𝐴𝐵 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 & ⊢ (𝐵 · 𝑃) = ;𝐸𝐷 ⇒ ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 | ||
6-Sep-2021 | decmul1 11461 | The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 = ;𝐴𝐵 & ⊢ 𝐷 ∈ ℕ0 & ⊢ (𝐴 · 𝑃) = 𝐶 & ⊢ (𝐵 · 𝑃) = 𝐷 ⇒ ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 | ||
6-Sep-2021 | decsubi 11459 | Difference between a numeral 𝑀 and a nonnegative integer 𝑁 (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ (𝐴 + 1) = 𝐷 & ⊢ (𝐵 − 𝑁) = 𝐶 ⇒ ⊢ (𝑀 − 𝑁) = ;𝐴𝐶 | ||
6-Sep-2021 | decaddci2 11457 | Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ (𝐴 + 1) = 𝐷 & ⊢ (𝐵 + 𝑁) = ;10 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐷0 | ||
6-Sep-2021 | decaddc2 11451 | Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 & ⊢ (𝐵 + 𝐷) = ;10 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐸0 | ||
6-Sep-2021 | decaddc 11448 | Add two numerals 𝑀 and 𝑁 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ ((𝐴 + 𝐶) + 1) = 𝐸 & ⊢ 𝐹 ∈ ℕ0 & ⊢ (𝐵 + 𝐷) = ;1𝐹 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐸𝐹 | ||
6-Sep-2021 | decadd 11446 | Add two numerals 𝑀 and 𝑁 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ (𝐴 + 𝐶) = 𝐸 & ⊢ (𝐵 + 𝐷) = 𝐹 ⇒ ⊢ (𝑀 + 𝑁) = ;𝐸𝐹 | ||
6-Sep-2021 | decma2c 11444 | Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplier 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐺 ∈ ℕ0 & ⊢ ((𝑃 · 𝐴) + (𝐶 + 𝐺)) = 𝐸 & ⊢ ((𝑃 · 𝐵) + 𝐷) = ;𝐺𝐹 ⇒ ⊢ ((𝑃 · 𝑀) + 𝑁) = ;𝐸𝐹 | ||
6-Sep-2021 | decmac 11442 | Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐺 ∈ ℕ0 & ⊢ ((𝐴 · 𝑃) + (𝐶 + 𝐺)) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝐷) = ;𝐺𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 | ||
6-Sep-2021 | decma 11440 | Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ 𝑁 = ;𝐶𝐷 & ⊢ 𝑃 ∈ ℕ0 & ⊢ ((𝐴 · 𝑃) + 𝐶) = 𝐸 & ⊢ ((𝐵 · 𝑃) + 𝐷) = 𝐹 ⇒ ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 | ||
6-Sep-2021 | dec10p 11429 | Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (;10 + 𝐴) = ;1𝐴 | ||
6-Sep-2021 | decsucc 11426 | The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ (𝐴 + 1) = 𝐵 & ⊢ 𝑁 = ;𝐴9 ⇒ ⊢ (𝑁 + 1) = ;𝐵0 | ||
6-Sep-2021 | declti 11422 | Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐶 < ;10 ⇒ ⊢ 𝐶 < ;𝐴𝐵 | ||
6-Sep-2021 | 3decltc 11414 | Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 15-Jun-2021.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐴 < 𝐵 & ⊢ 𝐶 < ;10 & ⊢ 𝐸 < ;10 ⇒ ⊢ ;;𝐴𝐶𝐸 < ;;𝐵𝐷𝐹 | ||
6-Sep-2021 | decsuc 11411 | The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ (𝐵 + 1) = 𝐶 & ⊢ 𝑁 = ;𝐴𝐵 ⇒ ⊢ (𝑁 + 1) = ;𝐴𝐶 | ||
6-Sep-2021 | decltc 11408 | Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐶 < ;10 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ;𝐴𝐶 < ;𝐵𝐷 | ||
6-Sep-2021 | declt 11406 | Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ & ⊢ 𝐵 < 𝐶 ⇒ ⊢ ;𝐴𝐵 < ;𝐴𝐶 | ||
6-Sep-2021 | decnncl2 11401 | Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ ⇒ ⊢ ;𝐴0 ∈ ℕ | ||
6-Sep-2021 | dec0h 11398 | Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ 𝐴 = ;0𝐴 | ||
6-Sep-2021 | dec0u 11396 | Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ (;10 · 𝐴) = ;𝐴0 | ||
6-Sep-2021 | decnncl 11394 | Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ ⇒ ⊢ ;𝐴𝐵 ∈ ℕ | ||
6-Sep-2021 | 10nn0 11392 | 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ ;10 ∈ ℕ0 | ||
6-Sep-2021 | 10nn 11390 | 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.) |
⊢ ;10 ∈ ℕ | ||
6-Sep-2021 | deccl 11388 | Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 ⇒ ⊢ ;𝐴𝐵 ∈ ℕ0 | ||
6-Sep-2021 | deceq2 11378 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵) | ||
6-Sep-2021 | deceq1 11376 | Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶) | ||
6-Sep-2021 | decex 11374 | A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
⊢ ;𝐴𝐵 ∈ V | ||
6-Sep-2021 | 9m1e8 11020 | 9 - 1 = 8. (Contributed by AV, 6-Sep-2021.) |
⊢ (9 − 1) = 8 | ||
6-Sep-2021 | 8m1e7 11019 | 8 - 1 = 7. (Contributed by AV, 6-Sep-2021.) |
⊢ (8 − 1) = 7 | ||
6-Sep-2021 | 7m1e6 11018 | 7 - 1 = 6. (Contributed by AV, 6-Sep-2021.) |
⊢ (7 − 1) = 6 | ||
6-Sep-2021 | 6m1e5 11017 | 6 - 1 = 5. (Contributed by AV, 6-Sep-2021.) |
⊢ (6 − 1) = 5 | ||
6-Sep-2021 | 5m1e4 11016 | 5 - 1 = 4. (Contributed by AV, 6-Sep-2021.) |
⊢ (5 − 1) = 4 | ||
6-Sep-2021 | 4m1e3 11015 | 4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.) |
⊢ (4 − 1) = 3 | ||
6-Sep-2021 | 3m1e2 11014 | 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.) (Proof shortened by AV, 6-Sep-2021.) |
⊢ (3 − 1) = 2 | ||
5-Sep-2021 | ex-lcm 26707 | Example for df-lcm 15141. (Contributed by AV, 5-Sep-2021.) |
⊢ (6 lcm 9) = ;18 | ||
5-Sep-2021 | ex-gcd 26706 | Example for df-gcd 15055. (Contributed by AV, 5-Sep-2021.) |
⊢ (-6 gcd 9) = 3 | ||
4-Sep-2021 | ex-abs 26704 | Example for df-abs 13824. (Contributed by AV, 4-Sep-2021.) |
⊢ (abs‘-2) = 2 | ||
4-Sep-2021 | ex-sqrt 26703 | Example for df-sqrt 13823. (Contributed by AV, 4-Sep-2021.) |
⊢ (√‘;25) = 5 | ||
4-Sep-2021 | ex-hash 26702 | Example for df-hash 12980. (Contributed by AV, 4-Sep-2021.) |
⊢ (#‘{0, 1, 2}) = 3 | ||
4-Sep-2021 | ex-bc 26701 | Example for df-bc 12952. (Contributed by AV, 4-Sep-2021.) |
⊢ (5C3) = ;10 | ||
4-Sep-2021 | ex-fac 26700 | Example for df-fac 12923. (Contributed by AV, 4-Sep-2021.) |
⊢ (!‘5) = ;;120 | ||
4-Sep-2021 | ex-exp 26699 | Example for df-exp 12723. (Contributed by AV, 4-Sep-2021.) |
⊢ ((5↑2) = ;25 ∧ (-3↑-2) = (1 / 9)) | ||
4-Sep-2021 | ex-ceil 26697 | Example for df-ceil 12456. (Contributed by AV, 4-Sep-2021.) |
⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1) | ||
4-Sep-2021 | 5t2e10 11510 | 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.) |
⊢ (5 · 2) = ;10 | ||
3-Sep-2021 | ex-mod 26698 | Example for df-mod 12531. (Contributed by AV, 3-Sep-2021.) |
⊢ ((5 mod 3) = 2 ∧ (-7 mod 2) = 1) | ||
2-Sep-2021 | setrec2 42241 |
This is the second of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(𝐹) is a subclass of all classes 𝐶 that
are closed
under 𝐹. Taken together, theorems setrec1 42237 and setrec2v 42242
uniquely determine setrecs(𝐹) to be the minimal class closed
under 𝐹.
We express this by saying that if 𝐹 respects the ⊆ relation and 𝐶 is closed under 𝐹, then 𝐵 ⊆ 𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 6945) to the other class. (Contributed by Emmett Weisz, 2-Sep-2021.) |
⊢ Ⅎ𝑎𝐹 & ⊢ 𝐵 = setrecs(𝐹) & ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | ||
1-Sep-2021 | grpidinv 17298 | A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (Revised by AV, 1-Sep-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → ∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑢 ∧ (𝑥 + 𝑦) = 𝑢))) | ||
1-Sep-2021 | grpidinv2 17297 | A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by AV, 1-Sep-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ((( 0 + 𝐴) = 𝐴 ∧ (𝐴 + 0 ) = 𝐴) ∧ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 ))) | ||
1-Sep-2021 | grplrinv 17296 | In a group, every member has a left and right inverse. (Contributed by AV, 1-Sep-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 )) | ||
31-Aug-2021 | mulginvinv 17389 | The group multiple operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘(𝑁 · (𝐼‘𝑋))) = (𝑁 · 𝑋)) | ||
31-Aug-2021 | mulginvcom 17388 | The group multiple operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · (𝐼‘𝑋)) = (𝐼‘(𝑁 · 𝑋))) | ||
31-Aug-2021 | mulgaddcom 17387 | The group multiple operator commutes with the group operation. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((𝑁 · 𝑋) + 𝑋) = (𝑋 + (𝑁 · 𝑋))) | ||
31-Aug-2021 | mulgaddcomlem 17386 | Lemma for mulgaddcom 17387. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 31-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋))) → ((-𝑦 · 𝑋) + 𝑋) = (𝑋 + (-𝑦 · 𝑋))) | ||
30-Aug-2021 | wl-cbv3vv 32486 | Avoiding ax-11 2021. (Contributed by Wolf Lammen, 30-Aug-2021.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
30-Aug-2021 | mulgmodid 17404 | Casting out multiples of the identity element leaves the group multiple unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ (𝑋 ∈ 𝐵 ∧ (𝑀 · 𝑋) = 0 )) → ((𝑁 mod 𝑀) · 𝑋) = (𝑁 · 𝑋)) | ||
30-Aug-2021 | mulgassr 17403 | Reversed product of group multiples. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑁 · 𝑀) · 𝑋) = (𝑀 · (𝑁 · 𝑋))) | ||
30-Aug-2021 | mulgnegneg 17384 | The inverse of a negative group multiple is the positive group multiple. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘(-𝑁 · 𝑋)) = (𝑁 · 𝑋)) | ||
30-Aug-2021 | grpasscan2 17302 | An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋) | ||
30-Aug-2021 | grpasscan1 17301 | An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 30-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((𝑁‘𝑋) + 𝑌)) = 𝑌) | ||
30-Aug-2021 | resgrpplusfrn 17259 | The underlying set of a group operation which is a restriction of a structure. (Contributed by Paul Chapman, 25-Mar-2008.) (Revised by AV, 30-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝑆) & ⊢ 𝐹 = (+𝑓‘𝐻) ⇒ ⊢ ((𝐻 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → 𝑆 = ran 𝐹) | ||
30-Aug-2021 | grpplusfo 17258 | The group addition operation is a function onto the base set/set of group elements. (Contributed by NM, 30-Oct-2006.) (Revised by AV, 30-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐹 = (+𝑓‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → 𝐹:(𝐵 × 𝐵)–onto→𝐵) | ||
30-Aug-2021 | issstrmgm 17075 | Characterize a substructure as submagma by closure properties. (Contributed by AV, 30-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑆 ⊆ 𝐵) → (𝐻 ∈ Mgm ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)) | ||
29-Aug-2021 | mulgnnass 17399 | Product of group multiples, for positive multiples in a semigroup. (Contributed by Mario Carneiro, 13-Dec-2014.) (Revised by AV, 29-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ SGrp ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 · 𝑁) · 𝑋) = (𝑀 · (𝑁 · 𝑋))) | ||
29-Aug-2021 | mulgnndir 17392 | Sum of group multiples, for positive multiples. (Contributed by Mario Carneiro, 11-Dec-2014.) (Revised by AV, 29-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ SGrp ∧ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵)) → ((𝑀 + 𝑁) · 𝑋) = ((𝑀 · 𝑋) + (𝑁 · 𝑋))) | ||
29-Aug-2021 | mulgnncl 17379 | Closure of the group multiple (exponentiation) operation for a positive multiplier in a magma. (Contributed by Mario Carneiro, 11-Dec-2014.) (Revised by AV, 29-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Mgm ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) ∈ 𝐵) | ||
28-Aug-2021 | elpglem3 42255 | Lemma for elpg 42256. (Contributed by Emmett Weisz, 28-Aug-2021.) |
⊢ (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))) | ||
28-Aug-2021 | elpglem2 42254 | Lemma for elpg 42256. (Contributed by Emmett Weisz, 28-Aug-2021.) |
⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))) | ||
28-Aug-2021 | elpglem1 42253 | Lemma for elpg 42256. (Contributed by Emmett Weisz, 28-Aug-2021.) |
⊢ (∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)) → ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)) | ||
28-Aug-2021 | dfgrp3e 17338 | Alternate definition of a group as a set with a closed, associative operation, for which solutions 𝑥 and 𝑦 of the equations (𝑎 + 𝑥) = 𝑏 and (𝑥 + 𝑎) = 𝑏 exist. Exercise 1 of [Herstein] p. 57. (Contributed by NM, 5-Dec-2006.) (Revised by AV, 28-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp ↔ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))) | ||
28-Aug-2021 | dfgrp3 17337 | Alternate definition of a group as semigroup (with at least one element) which is also a quasigroup, i.e. a magma in which solutions 𝑥 and 𝑦 of the equations (𝑎 + 𝑥) = 𝑏 and (𝑥 + 𝑎) = 𝑏 exist. Theorem 3.2 of [Bruck] p. 28. (Contributed by AV, 28-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))) | ||
28-Aug-2021 | dfgrp3lem 17336 | Lemma for dfgrp3 17337. (Contributed by AV, 28-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑢)) | ||
28-Aug-2021 | dfgrp2e 17271 | Alternate definition of a group as a set with a closed, associative operation, a left identity and a left inverse for each element. Alternate definition in [Lang] p. 7. (Contributed by NM, 10-Oct-2006.) (Revised by AV, 28-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛))) | ||
28-Aug-2021 | dfgrp2 17270 | Alternate definition of a group as semigroup with a left identity and a left inverse for each element. This "definition" is weaker than df-grp 17248, based on the definition of a monoid which provides a left and a right identity. (Contributed by AV, 28-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ SGrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛))) | ||
28-Aug-2021 | grpsgrp 17269 | A group is a semigroup. (Contributed by AV, 28-Aug-2021.) |
⊢ (𝐺 ∈ Grp → 𝐺 ∈ SGrp) | ||
28-Aug-2021 | sgrp0b 17115 | The structure with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.) |
⊢ {〈(Base‘ndx), ∅〉, 〈(+g‘ndx), 𝑂〉} ∈ SGrp | ||
28-Aug-2021 | sgrp0 17114 | Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.) |
⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ SGrp) | ||
28-Aug-2021 | mgm0b 17079 | The structure with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021.) |
⊢ {〈(Base‘ndx), ∅〉, 〈(+g‘ndx), 𝑂〉} ∈ Mgm | ||
28-Aug-2021 | mgm0 17078 | Any set with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021.) |
⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Mgm) | ||
27-Aug-2021 | elpg 42256 | Membership in the class of partizan games. In ONAG this is stated as "If 𝐿 and 𝑅 are any two sets of games, then there is a game {𝐿 ∣ 𝑅}. All games are constructed in this way." The first sentence corresponds to the backward direction of our theorem, and the second to the forward direction. (Contributed by Emmett Weisz, 27-Aug-2021.) |
⊢ (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)) | ||
27-Aug-2021 | hhssabloi 27503 | Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (Proof shortened by AV, 27-Aug-2021.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Sℋ ⇒ ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp | ||
27-Aug-2021 | hhssabloilem 27502 | Lemma for hhssabloi 27503. Formerly part of proof for hhssabloi 27503 which was based on the deprecated definition "SubGrpOp" for subgroups. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (Revised by AV, 27-Aug-2021.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Sℋ ⇒ ⊢ ( +ℎ ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ) | ||
27-Aug-2021 | ablnnncan 18051 | Cancellation law for group subtraction. (nnncan 10195 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − 𝑌)) | ||
27-Aug-2021 | csbprc 3932 | The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.) (Proof shortened by JJ, 27-Aug-2021.) |
⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) | ||
26-Aug-2021 | cnndv 31700 | There exists a continuous nowhere differentiable function. The result follows directly from knoppcn 31664 and knoppndv 31695. (Contributed by Asger C. Ipsen, 26-Aug-2021.) |
⊢ ∃𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅) | ||
26-Aug-2021 | cnndvlem2 31699 | Lemma for cnndv 31700. (Contributed by Asger C. Ipsen, 26-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) ⇒ ⊢ ∃𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅) | ||
26-Aug-2021 | cnmgpid 19627 | The group identity element of nonzero complex number multiplication is one. (Contributed by Steve Rodriguez, 23-Feb-2007.) (Revised by AV, 26-Aug-2021.) |
⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ⇒ ⊢ (0g‘𝑀) = 1 | ||
26-Aug-2021 | cnaddinv 18097 | Value of the group inverse of complex number addition. See also cnfldneg 19591. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 26-Aug-2021.) (New usage is discouraged.) |
⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ⇒ ⊢ (𝐴 ∈ ℂ → ((invg‘𝐺)‘𝐴) = -𝐴) | ||
26-Aug-2021 | cnaddid 18096 | The group identity element of complex number addition is zero. See also cnfld0 19589. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 26-Aug-2021.) (New usage is discouraged.) |
⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ⇒ ⊢ (0g‘𝐺) = 0 | ||
26-Aug-2021 | ghmghmrn 17502 | A group homomorphism from 𝐺 to 𝐻 is also a group homomorphism from 𝐺 to its image in 𝐻. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by AV, 26-Aug-2021.) |
⊢ 𝑈 = (𝑇 ↾s ran 𝐹) ⇒ ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑈)) | ||
26-Aug-2021 | grp1inv 17346 | The inverse function of the trivial group. (Contributed by FL, 21-Jun-2010.) (Revised by AV, 26-Aug-2021.) |
⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} ⇒ ⊢ (𝐼 ∈ 𝑉 → (invg‘𝑀) = ( I ↾ {𝐼})) | ||
25-Aug-2021 | cnndvlem1 31698 | Lemma for cnndv 31700. (Contributed by Asger C. Ipsen, 25-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) ⇒ ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅) | ||
25-Aug-2021 | knoppcn2 31697 | Variant of knoppcn 31664 with different codomain. (Contributed by Asger C. Ipsen, 25-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) ⇒ ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℝ)) | ||
25-Aug-2021 | knoppf 31696 | Knopp's function is a function. (Contributed by Asger C. Ipsen, 25-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝑊:ℝ⟶ℝ) | ||
25-Aug-2021 | restidsing 5377 | Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.) |
⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) | ||
25-Aug-2021 | resima2 5352 | Image under a restricted class. (Contributed by FL, 31-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.) |
⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵)) | ||
25-Aug-2021 | resiun1 5336 | Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) (Proof shortened by JJ, 25-Aug-2021.) |
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) | ||
24-Aug-2021 | ringinvnzdiv 18416 | In a unitary ring, a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.) (Revised by Jeff Madsen, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 (𝑎 · 𝑋) = 1 ) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑌 = 0 )) | ||
24-Aug-2021 | ringinvnz1ne0 18415 | In a unitary ring, a left invertible element is different from zero iff 1 ≠ 0. (Contributed by FL, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 (𝑎 · 𝑋) = 1 ) ⇒ ⊢ (𝜑 → (𝑋 ≠ 0 ↔ 1 ≠ 0 )) | ||
24-Aug-2021 | rngo2times 18399 | A ring element plus itself is two times the element. "Two" in an arbitrary unital ring is the sum of the unit with itself. (Contributed by AV, 24-Aug-2021.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵) → (𝐴 + 𝐴) = (( 1 + 1 ) · 𝐴)) | ||
24-Aug-2021 | ringadd2 18398 | A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ∃𝑥 ∈ 𝐵 (𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋)) | ||
24-Aug-2021 | ringid 18397 | The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ∃𝑢 ∈ 𝐵 ((𝑢 · 𝑋) = 𝑋 ∧ (𝑋 · 𝑢) = 𝑋)) | ||
24-Aug-2021 | srgfcl 18338 | Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by AV, 24-Aug-2021.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → · :(𝐵 × 𝐵)⟶𝐵) | ||
22-Aug-2021 | df-pg 42252 | Define the class of partizan games. More precisely, this is the class of partizan game forms, many of which represent equal partisan games. In metamath, equality between partizan games is represented by a different equivalence relation than class equality. (Contributed by Emmett Weisz, 22-Aug-2021.) |
⊢ Pg = setrecs((𝑥 ∈ V ↦ (𝒫 𝑥 × 𝒫 𝑥))) | ||
21-Aug-2021 | oddprmuzge3 40163 | A prime number which is odd is an integer greater than or equal to 3. (Contributed by AV, 20-Jul-2020.) (Proof shortened by AV, 21-Aug-2021.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∈ Odd ) → 𝑃 ∈ (ℤ≥‘3)) | ||
21-Aug-2021 | oddprmne2 40162 | Every prime number not being 2 is an odd prime number. (Contributed by AV, 21-Aug-2021.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∈ Odd ) ↔ 𝑃 ∈ (ℙ ∖ {2})) | ||
21-Aug-2021 | divalgmodcl 14969 | The result of the mod operator satisfies the requirements for the remainder 𝑅 in the division algorithm for a positive divisor. Variant of divalgmod 14967. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by AV, 21-Aug-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ 𝑅 ∈ ℕ0) → (𝑅 = (𝑁 mod 𝐷) ↔ (𝑅 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑅)))) | ||
21-Aug-2021 | divalgmod 14967 | The result of the mod operator satisfies the requirements for the remainder 𝑅 in the division algorithm for a positive divisor (compare divalg2 14966 and divalgb 14965). This demonstration theorem justifies the use of mod to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by AV, 21-Aug-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → (𝑅 = (𝑁 mod 𝐷) ↔ (𝑅 ∈ ℕ0 ∧ (𝑅 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑅))))) | ||
20-Aug-2021 | sgprmdvdsmersenne 40059 | If 𝑃 is a Sophie Germain prime (i.e. 𝑄 = ((2 · 𝑃) + 1) is also prime) with 𝑃≡3 (mod 4), then 𝑄 divides the 𝑃-th Mersenne number MP. (Contributed by AV, 20-Aug-2021.) |
⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 3) ∧ (𝑄 = ((2 · 𝑃) + 1) ∧ 𝑄 ∈ ℙ)) → 𝑄 ∥ ((2↑𝑃) − 1)) | ||
20-Aug-2021 | sfprmdvdsmersenne 40058 | If 𝑄 is a safe prime (i.e. 𝑄 = ((2 · 𝑃) + 1) for a prime 𝑃) with 𝑄≡7 (mod 8), then 𝑄 divides the 𝑃-th Mersenne number MP. (Contributed by AV, 20-Aug-2021.) |
⊢ ((𝑃 ∈ ℙ ∧ (𝑄 ∈ ℙ ∧ (𝑄 mod 8) = 7 ∧ 𝑄 = ((2 · 𝑃) + 1))) → 𝑄 ∥ ((2↑𝑃) − 1)) | ||
20-Aug-2021 | lgsqrmodndvds 24878 | If the Legendre symbol of an integer 𝐴 for an odd prime is 1, then the number is a quadratic residue mod 𝑃 with a solution 𝑥 of the congruence (𝑥↑2)≡𝐴 (mod 𝑃) which is not divisible by the prime. (Contributed by AV, 20-Aug-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) = 1 → ∃𝑥 ∈ ℤ (((𝑥↑2) mod 𝑃) = (𝐴 mod 𝑃) ∧ ¬ 𝑃 ∥ 𝑥))) | ||
20-Aug-2021 | lgsqrmod 24877 | If the Legendre symbol of an integer for an odd prime is 1, then the number is a quadratic residue mod 𝑃. (Contributed by AV, 20-Aug-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) = 1 → ∃𝑥 ∈ ℤ ((𝑥↑2) mod 𝑃) = (𝐴 mod 𝑃))) | ||
20-Aug-2021 | prmn2uzge3OLD 15251 | Obsolete version of oddprmge3 15250 as of 20-Aug-2021. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Proof shortened by AV, 20-Aug-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ≠ 2) → 𝑃 ∈ (ℤ≥‘3)) | ||
20-Aug-2021 | oddprmge3 15250 | An odd prime is greater than or equal to 3. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 20-Aug-2021.) |
⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ≥‘3)) | ||
20-Aug-2021 | oddprmgt2 15249 | An odd prime is greater than 2. (Contributed by AV, 20-Aug-2021.) |
⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃) | ||
19-Aug-2021 | mod42tp1mod8 40057 | If a number is 3 modulo 4, twice the number plus 1 is 7 modulo 8. (Contributed by AV, 19-Aug-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 4) = 3) → (((2 · 𝑁) + 1) mod 8) = 7) | ||
19-Aug-2021 | pwm1geoserALT 40040 | The n-th power of a number decreased by 1 expressed by the finite geometric series 1 + 𝐴↑1 + 𝐴↑2 +... + 𝐴↑(𝑁 − 1). This alternate proof of pwm1geoser 14439 is not based on geoser 14438, but on pwdif 40039 and therefore shorter than the original proof. (Contributed by AV, 19-Aug-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘))) | ||
19-Aug-2021 | pwdif 40039 | The difference of two numbers to the same power is the difference of the two numbers multiplied with a finite sum. Generalization of subsq 12834. See Wikipedia "Fermat number", section "Other theorems about Fermat numbers", https://en.wikipedia.org/wiki/Fermat_number, 5-Aug-2021. (Contributed by AV, 6-Aug-2021.) (Revised by AV, 19-Aug-2021.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑𝑁) − (𝐵↑𝑁)) = ((𝐴 − 𝐵) · Σ𝑘 ∈ (0..^𝑁)((𝐴↑𝑘) · (𝐵↑((𝑁 − 𝑘) − 1))))) | ||
19-Aug-2021 | knoppndv 31695 | The continuous nowhere differentiable function 𝑊 ( Knopp, K. (1918). Math. Z. 2, 1-26 ) is, in fact, nowhere differentiable. (Contributed by Asger C. Ipsen, 19-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → dom (ℝ D 𝑊) = ∅) | ||
19-Aug-2021 | knoppndvlem22 31694 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 19-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐻 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎 ≤ 𝐻 ∧ 𝐻 ≤ 𝑏) ∧ ((𝑏 − 𝑎) < 𝐷 ∧ 𝑎 ≠ 𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊‘𝑏) − (𝑊‘𝑎))) / (𝑏 − 𝑎)))) | ||
19-Aug-2021 | modremain 14970 | The result of the modulo operation is the remainder of the division algorithm. (Contributed by AV, 19-Aug-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → ((𝑁 mod 𝐷) = 𝑅 ↔ ∃𝑧 ∈ ℤ ((𝑧 · 𝐷) + 𝑅) = 𝑁)) | ||
18-Aug-2021 | m11nprm 40056 | The eleventh Mersenne number M11 = 2047 is not a prime number. (Contributed by AV, 18-Aug-2021.) |
⊢ ((2↑;11) − 1) = (;89 · ;23) | ||
18-Aug-2021 | m7prm 40054 | The seventh Mersenne number M7 = 127 is a prime number. (Contributed by AV, 18-Aug-2021.) |
⊢ ((2↑7) − 1) ∈ ℙ | ||
18-Aug-2021 | 139prmALT 40049 | 139 is a prime number. In contrast to 139prm 15669, the proof of this theorem uses 3dvds2dec 14894 for checking the divisibility by 3. Although the proof using 3dvds2dec 14894 is longer (regarding size: 1849 characters compared with 1809 for 139prm 15669), the number of essential steps is smaller (301 compared with 327 for 139prm 15669). (Contributed by Mario Carneiro, 19-Feb-2014.) (Revised by AV, 18-Aug-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ;;139 ∈ ℙ | ||
18-Aug-2021 | 3ndvds4 40048 | 3 does not divide 4. (Contributed by AV, 18-Aug-2021.) |
⊢ ¬ 3 ∥ 4 | ||
18-Aug-2021 | knoppndvlem21 31693 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 18-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ 𝐺 = (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐻 ∈ ℝ) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) & ⊢ (𝜑 → (((2 · 𝑁)↑-𝐽) / 2) < 𝐷) & ⊢ (𝜑 → 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝐽) · 𝐺)) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎 ≤ 𝐻 ∧ 𝐻 ≤ 𝑏) ∧ ((𝑏 − 𝑎) < 𝐷 ∧ 𝑎 ≠ 𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊‘𝑏) − (𝑊‘𝑎))) / (𝑏 − 𝑎)))) | ||
18-Aug-2021 | knoppndvlem20 31692 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 18-Aug-2021.) |
⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+) | ||
18-Aug-2021 | o2p2e4 7508 | 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 5646. For the usual proof using complex numbers, see 2p2e4 11021. (Contributed by NM, 18-Aug-2021.) |
⊢ (2𝑜 +𝑜 2𝑜) = 4𝑜 | ||
17-Aug-2021 | m5prm 40051 | The fifth Mersenne number M5 = 31 is a prime number. (Contributed by AV, 17-Aug-2021.) |
⊢ ((2↑5) − 1) ∈ ℙ | ||
17-Aug-2021 | 31prm 40050 | 31 is a prime number. In contrast to 37prm 15666, the proof of this theorem is not based on the "blanket" prmlem2 15665, but on isprm7 15258. Although the checks for non-divisibility by the primes 7 to 23 are not needed, the proof is much longer (regarding size) than the proof of 37prm 15666 (1810 characters compared with 1213 for 37prm 15666). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 15666). (Contributed by AV, 17-Aug-2021.) (Proof modification is discouraged.) |
⊢ ;31 ∈ ℙ | ||
17-Aug-2021 | flsqrt5 40047 | The floor of the square root of a nonnegative number is 5 iff the number is between 25 and 35. (Contributed by AV, 17-Aug-2021.) |
⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((;25 ≤ 𝑋 ∧ 𝑋 < ;36) ↔ (⌊‘(√‘𝑋)) = 5)) | ||
17-Aug-2021 | flsqrt 40046 | A condition equivalent to the floor of a square root. (Contributed by AV, 17-Aug-2021.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℕ0) → ((⌊‘(√‘𝐴)) = 𝐵 ↔ ((𝐵↑2) ≤ 𝐴 ∧ 𝐴 < ((𝐵 + 1)↑2)))) | ||
17-Aug-2021 | knoppndvlem19 31691 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 17-Aug-2021.) |
⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑚) & ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑚 + 1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝐻 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ∃𝑚 ∈ ℤ (𝐴 ≤ 𝐻 ∧ 𝐻 ≤ 𝐵)) | ||
17-Aug-2021 | rpnnen2 14794 |
The other half of rpnnen 14795, where we show an injection from sets of
positive integers to real numbers. The obvious choice for this is
binary expansion, but it has the unfortunate property that it does not
produce an injection on numbers which end with all 0's or all 1's (the
more well-known decimal version of this is 0.999... 14451). Instead, we
opt for a ternary expansion, which produces (a scaled version of) the
Cantor set. Since the Cantor set is riddled with gaps, we can show that
any two sequences that are not equal must differ somewhere, and when
they do, they are placed a finite distance apart, thus ensuring that the
map is injective.
Our map assigns to each subset 𝐴 of the positive integers the number Σ𝑘 ∈ 𝐴(3↑-𝑘) = Σ𝑘 ∈ ℕ((𝐹‘𝐴)‘𝑘), where ((𝐹‘𝐴)‘𝑘) = if(𝑘 ∈ 𝐴, (3↑-𝑘), 0)) (rpnnen2lem1 14782). This is an infinite sum of real numbers (rpnnen2lem2 14783), and since 𝐴 ⊆ 𝐵 implies (𝐹‘𝐴) ≤ (𝐹‘𝐵) (rpnnen2lem4 14785) and (𝐹‘ℕ) converges to 1 / 2 (rpnnen2lem3 14784) by geoisum1 14449, the sum is convergent to some real (rpnnen2lem5 14786 and rpnnen2lem6 14787) by the comparison test for convergence cvgcmp 14389. The comparison test also tells us that 𝐴 ⊆ 𝐵 implies Σ(𝐹‘𝐴) ≤ Σ(𝐹‘𝐵) (rpnnen2lem7 14788). Putting it all together, if we have two sets 𝑥 ≠ 𝑦, there must differ somewhere, and so there must be an 𝑚 such that ∀𝑛 < 𝑚(𝑛 ∈ 𝑥 ↔ 𝑛 ∈ 𝑦) but 𝑚 ∈ (𝑥 ∖ 𝑦) or vice versa. In this case, we split off the first 𝑚 − 1 terms (rpnnen2lem8 14789) and cancel them (rpnnen2lem10 14791), since these are the same for both sets. For the remaining terms, we use the subset property to establish that Σ(𝐹‘𝑦) ≤ Σ(𝐹‘(ℕ ∖ {𝑚})) and Σ(𝐹‘{𝑚}) ≤ Σ(𝐹‘𝑥) (where these sums are only over (ℤ≥‘𝑚)), and since Σ(𝐹‘(ℕ ∖ {𝑚})) = (3↑-𝑚) / 2 (rpnnen2lem9 14790) and Σ(𝐹‘{𝑚}) = (3↑-𝑚), we establish that Σ(𝐹‘𝑦) < Σ(𝐹‘𝑥) (rpnnen2lem11 14792) so that they must be different. By contraposition (rpnnen2lem12 14793), we find that this map is an injection. (Contributed by Mario Carneiro, 13-May-2013.) (Proof shortened by Mario Carneiro, 30-Apr-2014.) (Revised by NM, 17-Aug-2021.) |
⊢ 𝒫 ℕ ≼ (0[,]1) | ||
17-Aug-2021 | declecOLD 11420 | Obsolete version of decleh 11417 as of 8-Sep-2021. (Contributed by AV, 17-Aug-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐶 ≤ 9 & ⊢ 𝐴 < 𝐵 ⇒ ⊢ ;𝐴𝐶 ≤ ;𝐵𝐷 | ||
17-Aug-2021 | decleOLD 11419 | Obsolete version of decle 11416 as of 8-Sep-2021. (Contributed by AV, 17-Aug-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐵 ≤ 𝐶 ⇒ ⊢ ;𝐴𝐵 ≤ ;𝐴𝐶 | ||
17-Aug-2021 | declei 11418 | Comparing a digit to a decimal integer. (Contributed by AV, 17-Aug-2021.) |
⊢ 𝐴 ∈ ℕ & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐶 ≤ 9 ⇒ ⊢ 𝐶 ≤ ;𝐴𝐵 | ||
16-Aug-2021 | lighneal 40066 | If a power of a prime 𝑃 (i.e. 𝑃↑𝑀) is of the form 2↑𝑁 − 1, then 𝑁 must be prime and 𝑀 must be 1. Generalization of mersenne 24752 (where 𝑀 = 1 is a prerequisite). Theorem of S. Ligh and L. Neal (1974) "A note on Mersenne mumbers", Mathematics Magazine, 47:4, 231-233. (Contributed by AV, 16-Aug-2021.) |
⊢ (((𝑃 ∈ ℙ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → (𝑀 = 1 ∧ 𝑁 ∈ ℙ)) | ||
16-Aug-2021 | lighneallem4 40065 | Lemma 3 for lighneal 40066. (Contributed by AV, 16-Aug-2021.) |
⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → 𝑀 = 1) | ||
16-Aug-2021 | lighneallem4b 40064 | Lemma 2 for lighneallem4 40065. (Contributed by AV, 16-Aug-2021.) |
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ (ℤ≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ (ℤ≥‘2)) | ||
16-Aug-2021 | lighneallem4a 40063 | Lemma 1 for lighneallem4 40065. (Contributed by AV, 16-Aug-2021.) |
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ (ℤ≥‘3) ∧ 𝑆 = (((𝐴↑𝑀) + 1) / (𝐴 + 1))) → 2 ≤ 𝑆) | ||
16-Aug-2021 | 2exp11 40055 | Two to the eleventh power is 2048. (Contributed by AV, 16-Aug-2021.) |
⊢ (2↑;11) = ;;;2048 | ||
16-Aug-2021 | 2exp7 40052 | Two to the seventh power is 128. (Contributed by AV, 16-Aug-2021.) |
⊢ (2↑7) = ;;128 | ||
16-Aug-2021 | 2exp5 40045 | Two to the fifth power is 32. (Contributed by AV, 16-Aug-2021.) |
⊢ (2↑5) = ;32 | ||
16-Aug-2021 | m3prm 40044 | The third Mersenne number M3 = 7 is a prime number. (Contributed by AV, 16-Aug-2021.) |
⊢ ((2↑3) − 1) ∈ ℙ | ||
16-Aug-2021 | m2prm 40043 | The second Mersenne number M2 = 3 is a prime number. (Contributed by AV, 16-Aug-2021.) |
⊢ ((2↑2) − 1) ∈ ℙ | ||
16-Aug-2021 | oddge22np1 14911 | An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → (¬ 2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ ((2 · 𝑛) + 1) = 𝑁)) | ||
15-Aug-2021 | oddpwp1fsum 14953 | An odd power of a number increased by 1 expressed by a product with a finite sum. (Contributed by AV, 15-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → ¬ 2 ∥ 𝑁) ⇒ ⊢ (𝜑 → ((𝐴↑𝑁) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑁 − 1))((-1↑𝑘) · (𝐴↑𝑘)))) | ||
15-Aug-2021 | pwp1fsum 14952 | The n-th power of a number increased by 1 expressed by a product with a finite sum. (Contributed by AV, 15-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (((-1↑(𝑁 − 1)) · (𝐴↑𝑁)) + 1) = ((𝐴 + 1) · Σ𝑘 ∈ (0...(𝑁 − 1))((-1↑𝑘) · (𝐴↑𝑘)))) | ||
15-Aug-2021 | n2dvdsm1 14943 | 2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.) |
⊢ ¬ 2 ∥ -1 | ||
15-Aug-2021 | rpnnen1 11696 | One half of rpnnen 14795, where we show an injection from the real numbers to sequences of rational numbers. Specifically, we map a real number 𝑥 to the sequence (𝐹‘𝑥):ℕ⟶ℚ (see rpnnen1lem6 11695) such that ((𝐹‘𝑥)‘𝑘) is the largest rational number with denominator 𝑘 that is strictly less than 𝑥. In this manner, we get a monotonically increasing sequence that converges to 𝑥, and since each sequence converges to a unique real number, this mapping from reals to sequences of rational numbers is injective. Note: The ℕ and ℚ existence hypotheses provide for use with either nnex 10903 and qex 11676, or nnexALT 10899 and qexALT 11679. The proof should not be modified to use any of those 4 theorems. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 16-Jun-2013.) (Revised by NM, 15-Aug-2021.) (Proof modification is discouraged.) |
⊢ ℕ ∈ V & ⊢ ℚ ∈ V ⇒ ⊢ ℝ ≼ (ℚ ↑𝑚 ℕ) | ||
15-Aug-2021 | rpnnen1lem6 11695 | Lemma for rpnnen1 11696. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 15-Aug-2021.) (Proof modification is discouraged.) |
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))) & ⊢ ℕ ∈ V & ⊢ ℚ ∈ V ⇒ ⊢ ℝ ≼ (ℚ ↑𝑚 ℕ) | ||
14-Aug-2021 | knoppndvlem18 31690 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 14-Aug-2021.) |
⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐺 ∈ ℝ+) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ ℕ0 ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · 𝐺))) | ||
14-Aug-2021 | oddsumodd 14951 | If every term in a sum with an odd number of terms is odd, then the sum is odd. (Contributed by AV, 14-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 2 ∥ 𝐵) & ⊢ (𝜑 → ¬ 2 ∥ (#‘𝐴)) ⇒ ⊢ (𝜑 → ¬ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵) | ||
14-Aug-2021 | evensumodd 14950 | If every term in a sum with an even number of terms is odd, then the sum is even. (Contributed by AV, 14-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 2 ∥ 𝐵) & ⊢ (𝜑 → 2 ∥ (#‘𝐴)) ⇒ ⊢ (𝜑 → 2 ∥ Σ𝑘 ∈ 𝐴 𝐵) | ||
14-Aug-2021 | sumodd 14949 | If every term in a sum is odd, then the sum is even iff the number of terms in the sum is even. (Contributed by AV, 14-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 2 ∥ 𝐵) ⇒ ⊢ (𝜑 → (2 ∥ (#‘𝐴) ↔ 2 ∥ Σ𝑘 ∈ 𝐴 𝐵)) | ||
14-Aug-2021 | sumeven 14948 | If every term in a sum is even, then so is the sum. (Contributed by AV, 14-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 2 ∥ 𝐵) ⇒ ⊢ (𝜑 → 2 ∥ Σ𝑘 ∈ 𝐴 𝐵) | ||
14-Aug-2021 | pwm1geoser 14439 | The n-th power of a number decreased by 1 expressed by the finite geometric series 1 + 𝐴↑1 + 𝐴↑2 +... + 𝐴↑(𝑁 − 1). (Contributed by AV, 14-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘))) | ||
13-Aug-2021 | lighneallem2 40061 | Lemma 2 for lighneal 40066. (Contributed by AV, 13-Aug-2021.) |
⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 2 ∥ 𝑁 ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → 𝑀 = 1) | ||
13-Aug-2021 | difsqpwdvds 15429 | If the difference of two squares is a power of a prime, the prime divides twice the second squared number. (Contributed by AV, 13-Aug-2021.) |
⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ (𝐵 + 1) < 𝐴) ∧ (𝐶 ∈ ℙ ∧ 𝐷 ∈ ℕ0)) → ((𝐶↑𝐷) = ((𝐴↑2) − (𝐵↑2)) → 𝐶 ∥ (2 · 𝐵))) | ||
13-Aug-2021 | dvdsprmpweqnn 15427 | If an integer greater than 1 divides a prime power, it is a (proper) prime power. (Contributed by AV, 13-Aug-2021.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 ∥ (𝑃↑𝑁) → ∃𝑛 ∈ ℕ 𝐴 = (𝑃↑𝑛))) | ||
13-Aug-2021 | rpnnen1lem5 11694 | Lemma for rpnnen1 11696. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.) |
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))) & ⊢ ℕ ∈ V & ⊢ ℚ ∈ V ⇒ ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥) | ||
13-Aug-2021 | rpnnen1lem4 11693 | Lemma for rpnnen1 11696. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.) |
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))) & ⊢ ℕ ∈ V & ⊢ ℚ ∈ V ⇒ ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ) | ||
13-Aug-2021 | rpnnen1lem3 11692 | Lemma for rpnnen1 11696. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.) |
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))) & ⊢ ℕ ∈ V & ⊢ ℚ ∈ V ⇒ ⊢ (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥) | ||
13-Aug-2021 | rpnnen1lem1 11691 | Lemma for rpnnen1 11696. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.) |
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))) & ⊢ ℕ ∈ V & ⊢ ℚ ∈ V ⇒ ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) ∈ (ℚ ↑𝑚 ℕ)) | ||
13-Aug-2021 | difgtsumgt 11223 | If the difference of a real number and a nonnegative integer is greater than another real number, the sum of the real number and the nonnegative integer is also greater than the other real number. (Contributed by AV, 13-Aug-2021.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐶 < (𝐴 − 𝐵) → 𝐶 < (𝐴 + 𝐵))) | ||
13-Aug-2021 | nn0negleid 11222 | A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.) |
⊢ (𝐴 ∈ ℕ0 → -𝐴 ≤ 𝐴) | ||
13-Aug-2021 | 0nelxp 5067 | The empty set is not a member of a Cartesian product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof shortened by JJ, 13-Aug-2021.) |
⊢ ¬ ∅ ∈ (𝐴 × 𝐵) | ||
13-Aug-2021 | elxp2 5056 | Membership in a Cartesian product. (Contributed by NM, 23-Feb-2004.) (Proof shortened by JJ, 13-Aug-2021.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉) | ||
12-Aug-2021 | knoppndvlem17 31689 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 12-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → ((((2 · 𝑁) · (abs‘𝐶))↑𝐽) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ ((abs‘((𝑊‘𝐵) − (𝑊‘𝐴))) / (𝐵 − 𝐴))) | ||
12-Aug-2021 | evennn2n 14913 | A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.) |
⊢ (𝑁 ∈ ℕ → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ (2 · 𝑛) = 𝑁)) | ||
12-Aug-2021 | evennn02n 14912 | A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.) |
⊢ (𝑁 ∈ ℕ0 → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ0 (2 · 𝑛) = 𝑁)) | ||
11-Aug-2021 | lighneallem3 40062 | Lemma 3 for lighneal 40066. (Contributed by AV, 11-Aug-2021.) |
⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → 𝑀 = 1) | ||
11-Aug-2021 | lighneallem1 40060 | Lemma 1 for lighneal 40066. (Contributed by AV, 11-Aug-2021.) |
⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2↑𝑁) − 1) ≠ (𝑃↑𝑀)) | ||
11-Aug-2021 | vtoclgft 3227 | Closed theorem form of vtoclgf 3237. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.) |
⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → 𝜓) | ||
9-Aug-2021 | dvdsdivcl 14876 | The complement of a divisor of 𝑁 is also a divisor of 𝑁. (Contributed by Mario Carneiro, 2-Jul-2015.) (Proof shortened by AV, 9-Aug-2021.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝐴) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) | ||
8-Aug-2021 | wl-hbae1 32482 | This specialization of hbae 2303 does not depend on ax-11 2021. (Contributed by Wolf Lammen, 8-Aug-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦) | ||
8-Aug-2021 | isoddgcd1 15277 | The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by AV, 8-Aug-2021.) |
⊢ (𝑍 ∈ ℤ → (¬ 2 ∥ 𝑍 ↔ (2 gcd 𝑍) = 1)) | ||
8-Aug-2021 | isevengcd2 15276 | The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) (Revised by AV, 8-Aug-2021.) |
⊢ (𝑍 ∈ ℤ → (2 ∥ 𝑍 ↔ (2 gcd 𝑍) = 2)) | ||
8-Aug-2021 | gcdeq 15110 | 𝐴 is equal to its gcd with 𝐵 if and only if 𝐴 divides 𝐵. (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by AV, 8-Aug-2021.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵) = 𝐴 ↔ 𝐴 ∥ 𝐵)) | ||
8-Aug-2021 | dfgcd2 15101 | Alternate definition of the gcd operator, see definition in [ApostolNT] p. 15. (Contributed by AV, 8-Aug-2021.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐷 = (𝑀 gcd 𝑁) ↔ (0 ≤ 𝐷 ∧ (𝐷 ∥ 𝑀 ∧ 𝐷 ∥ 𝑁) ∧ ∀𝑒 ∈ ℤ ((𝑒 ∥ 𝑀 ∧ 𝑒 ∥ 𝑁) → 𝑒 ∥ 𝐷)))) | ||
7-Aug-2021 | 2pwp1prmfmtno 40042 | Every prime number of the form ((2↑𝑘) + 1) must be a Fermat number. (Contributed by AV, 7-Aug-2021.) |
⊢ ((𝐾 ∈ ℕ ∧ 𝑃 = ((2↑𝐾) + 1) ∧ 𝑃 ∈ ℙ) → ∃𝑛 ∈ ℕ0 𝑃 = (FermatNo‘𝑛)) | ||
7-Aug-2021 | 2pwp1prm 40041 | For every prime number of the form ((2↑𝑘) + 1) 𝑘 must be a power of 2, see Wikipedia "Fermat number", section "Other theorms about Fermat numbers", https://en.wikipedia.org/wiki/Fermat_number, 5-Aug-2021. (Contributed by AV, 7-Aug-2021.) |
⊢ ((𝐾 ∈ ℕ ∧ ((2↑𝐾) + 1) ∈ ℙ) → ∃𝑛 ∈ ℕ0 𝐾 = (2↑𝑛)) | ||
7-Aug-2021 | divconjdvds 14875 | If a nonzero integer 𝑀 divides another integer 𝑁, the other integer 𝑁 divided by the nonzero integer 𝑀 (i.e. the divisor conjugate of 𝑁 to 𝑀) divides the other integer 𝑁. Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV, 7-Aug-2021.) |
⊢ ((𝑀 ∥ 𝑁 ∧ 𝑀 ≠ 0) → (𝑁 / 𝑀) ∥ 𝑁) | ||
7-Aug-2021 | dvdseq 14874 | If two nonnegative integers divide each other, they must be equal. (Contributed by Mario Carneiro, 30-May-2014.) (Proof shortened by AV, 7-Aug-2021.) |
⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀)) → 𝑀 = 𝑁) | ||
7-Aug-2021 | dvdsabseq 14873 | If two integers divide each other, they must be equal, up to a difference in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.) |
⊢ ((𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀) → (abs‘𝑀) = (abs‘𝑁)) | ||
7-Aug-2021 | ralxfrd 4805 | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒)) | ||
7-Aug-2021 | reusv2lem2 4795 | Lemma for reusv2 4800. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.) |
⊢ (∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵 → ∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵) | ||
7-Aug-2021 | reusv1 4792 | Two ways to express single-valuedness of a class expression 𝐶(𝑦). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.) |
⊢ (∃𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) | ||
6-Aug-2021 | oddprmdvds 15445 | Every positive integer which is not a power of two is divisible by an odd prime number. (Contributed by AV, 6-Aug-2021.) |
⊢ ((𝐾 ∈ ℕ ∧ ¬ ∃𝑛 ∈ ℕ0 𝐾 = (2↑𝑛)) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾) | ||
4-Aug-2021 | prminf2 40038 | The set of prime numbers is infinite. The proof of this variant of prminf 15457 is based on Goldbach's theorem goldbachth 39997 (via prmdvdsfmtnof1 40037 and prmdvdsfmtnof1lem2 40035), see Wikipedia "Fermat number", 4-Aug-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 4-Aug-2021.) |
⊢ ℙ ∉ Fin | ||
4-Aug-2021 | prmdvdsfmtnof1 40037 | The mapping of a Fermat number to its smallest prime factor is a one-to-one function. (Contributed by AV, 4-Aug-2021.) |
⊢ 𝐹 = (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < )) ⇒ ⊢ 𝐹:ran FermatNo–1-1→ℙ | ||
4-Aug-2021 | prmdvdsfmtnof 40036 | The mapping of a Fermat number to its smallest prime factor is a function. (Contributed by AV, 4-Aug-2021.) |
⊢ 𝐹 = (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < )) ⇒ ⊢ 𝐹:ran FermatNo⟶ℙ | ||
4-Aug-2021 | fmtnonn 39981 | Each Fermat number is a positive integer. (Contributed by AV, 26-Jul-2021.) (Proof shortened by AV, 4-Aug-2021.) |
⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℕ) | ||
4-Aug-2021 | fmtnoge3 39980 | Each Fermat number is greater than or equal to 3. (Contributed by AV, 4-Aug-2021.) |
⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ≥‘3)) | ||
4-Aug-2021 | coprmdvds1 15203 | If two positive integers are coprime, i.e. their greatest common divisor is 1, the only positive integer that divides both of them is 1. (Contributed by AV, 4-Aug-2021.) |
⊢ ((𝐹 ∈ ℕ ∧ 𝐺 ∈ ℕ ∧ (𝐹 gcd 𝐺) = 1) → ((𝐼 ∈ ℕ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐼 = 1)) | ||
3-Aug-2021 | prmdvdsfmtnof1lem2 40035 | Lemma 2 for prmdvdsfmtnof1 40037. (Contributed by AV, 3-Aug-2021.) |
⊢ ((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺)) | ||
3-Aug-2021 | prmdvdsfmtnof1lem1 40034 | Lemma 1 for prmdvdsfmtnof1 40037. (Contributed by AV, 3-Aug-2021.) |
⊢ 𝐼 = inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐹}, ℝ, < ) & ⊢ 𝐽 = inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐺}, ℝ, < ) ⇒ ⊢ ((𝐹 ∈ (ℤ≥‘2) ∧ 𝐺 ∈ (ℤ≥‘2)) → (𝐼 = 𝐽 → (𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺))) | ||
3-Aug-2021 | fmtnoinf 39986 | The set of Fermat numbers is infinite. (Contributed by AV, 3-Aug-2021.) |
⊢ ran FermatNo ∉ Fin | ||
3-Aug-2021 | fmtnof1 39985 | The enumeration of the Fermat numbers is a one-one function into the positive integers. (Contributed by AV, 3-Aug-2021.) |
⊢ FermatNo:ℕ0–1-1→ℕ | ||
3-Aug-2021 | fmtnorn 39984 | A Fermat number is a function value of the enumeration of the Fermat numbers. (Contributed by AV, 3-Aug-2021.) |
⊢ (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹) | ||
2-Aug-2021 | fmtnorec3 39998 | The third recurrence relation for Fermat numbers, see Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 2-Aug-2021.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → (FermatNo‘𝑁) = ((FermatNo‘(𝑁 − 1)) + ((2↑(2↑(𝑁 − 1))) · ∏𝑛 ∈ (0...(𝑁 − 2))(FermatNo‘𝑛)))) | ||
2-Aug-2021 | rmspecsqrtnq 36488 | The discriminant used to define the X and Y sequences has an irrational square root. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof shortened by AV, 2-Aug-2021.) |
⊢ (𝐴 ∈ (ℤ≥‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ)) | ||
2-Aug-2021 | loglesqrt 24299 | An upper bound on the logarithm. (Contributed by Mario Carneiro, 2-May-2016.) (Proof shortened by AV, 2-Aug-2021.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(𝐴 + 1)) ≤ (√‘𝐴)) | ||
2-Aug-2021 | fproddvdsd 14897 | A finite product of integers is divisible by any of its factors. (Contributed by AV, 14-Aug-2020.) (Proof shortened by AV, 2-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ ℤ) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∥ ∏𝑘 ∈ 𝐴 𝑘) | ||
2-Aug-2021 | fprodcom2 14553 | Interchange order of multiplication. Note that 𝐵(𝑗) and 𝐷(𝑘) are not necessarily constant expressions. (Contributed by Scott Fenton, 1-Feb-2018.) (Proof shortened by JJ, 2-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin) & ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))) & ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ) ⇒ ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ∏𝑘 ∈ 𝐵 𝐸 = ∏𝑘 ∈ 𝐶 ∏𝑗 ∈ 𝐷 𝐸) | ||
2-Aug-2021 | arisum2 14432 | Arithmetic series sum of the first 𝑁 nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 2-Aug-2021.) |
⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2)) | ||
2-Aug-2021 | fsumcom2 14347 | Interchange order of summation. Note that 𝐵(𝑗) and 𝐷(𝑘) are not necessarily constant expressions. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.) (Proof shortened by JJ, 2-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin) & ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐷))) & ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐸 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 𝐸 = Σ𝑘 ∈ 𝐶 Σ𝑗 ∈ 𝐷 𝐸) | ||
2-Aug-2021 | binom2sub1 12844 | Special case of binom2sub 12843 where 𝐵 = 1. (Contributed by AV, 2-Aug-2021.) |
⊢ (𝐴 ∈ ℂ → ((𝐴 − 1)↑2) = (((𝐴↑2) − (2 · 𝐴)) + 1)) | ||
2-Aug-2021 | fsuppmapnn0fiub 12652 | If all functions of a finite set of functions over the nonnegative integers are finitely supported, then the support of all these functions is contained in a finite set of sequential integers starting at 0 and ending with the supremum of the union of the support of these functions. (Contributed by AV, 2-Oct-2019.) (Proof shortened by JJ, 2-Aug-2021.) |
⊢ 𝑈 = ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) & ⊢ 𝑆 = sup(𝑈, ℝ, < ) ⇒ ⊢ ((𝑀 ⊆ (𝑅 ↑𝑚 ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → ((∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 ∧ 𝑈 ≠ ∅) → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑆))) | ||
1-Aug-2021 | goldbachth 39997 | Goldbach's theorem: Two different Fermat numbers are coprime. See ProofWiki "Goldbach's theorem", 31-Jul-2021, https://proofwiki.org/wiki/Goldbach%27s_Theorem or Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 1-Aug-2021.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ≠ 𝑀) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1) | ||
1-Aug-2021 | goldbachthlem2 39996 | Lemma 2 for goldbachth 39997. (Contributed by AV, 1-Aug-2021.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1) | ||
1-Aug-2021 | goldbachthlem1 39995 | Lemma 1 for goldbachth 39997. (Contributed by AV, 1-Aug-2021.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁) → (FermatNo‘𝑀) ∥ ((FermatNo‘𝑁) − 2)) | ||
1-Aug-2021 | fmtnodvds 39994 | Any Fermat number divides a greater Fermat number minus 2. Corrolary of fmtnorec2 39993, see ProofWiki "Product of Sequence of Fermat Numbers plus 2/Corollary", 31-Jul-2021. (Contributed by AV, 1-Aug-2021.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ) → (FermatNo‘𝑁) ∥ ((FermatNo‘(𝑁 + 𝑀)) − 2)) | ||
1-Aug-2021 | fprodfvdvdsd 14896 | A finite product of integers is divisible by any of its factors being function values. (Contributed by AV, 1-Aug-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐹:𝐵⟶ℤ) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∥ ∏𝑘 ∈ 𝐴 (𝐹‘𝑘)) | ||
1-Aug-2021 | 3dvds2dec 14894 | A decimal number is divisible by three iff the sum of its three "digits" is divisible by three. The term "digits" in its narrow sense is only correct if 𝐴, 𝐵 and 𝐶 actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers 𝐴, 𝐵 and 𝐶. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 ⇒ ⊢ (3 ∥ ;;𝐴𝐵𝐶 ↔ 3 ∥ ((𝐴 + 𝐵) + 𝐶)) | ||
1-Aug-2021 | 3dec 12912 | A "decimal constructor" which is used to build up "decimal integers" or "numeric terms" in base 10 with 3 "digits". (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 ⇒ ⊢ ;;𝐴𝐵𝐶 = ((((;10↑2) · 𝐴) + (;10 · 𝐵)) + 𝐶) | ||
1-Aug-2021 | sq10e99m1 12911 | The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
⊢ (;10↑2) = (;99 + 1) | ||
1-Aug-2021 | sq10 12910 | The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
⊢ (;10↑2) = ;;100 | ||
1-Aug-2021 | dfdec10 11373 | Version of the definition of the "decimal constructor" using ;10 instead of the symbol 10. Of course, this statement cannot be used as definition, because it uses the "decimal constructor". (Contributed by AV, 1-Aug-2021.) |
⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | ||
1-Aug-2021 | 9p1e10 11372 | 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.) |
⊢ (9 + 1) = ;10 | ||
1-Aug-2021 | df-dec 11370 | Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, (;;;1000 + ;;;2000) = ;;;3000 1kp2ke3k 26695. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.) |
⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) | ||
31-Jul-2021 | fmtnorec4 39999 | The fourth recurrence relation for Fermat numbers, see Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 31-Jul-2021.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → (FermatNo‘𝑁) = (((FermatNo‘(𝑁 − 1))↑2) − (2 · (((FermatNo‘(𝑁 − 2)) − 1)↑2)))) | ||
30-Jul-2021 | fmtnofac1 40020 |
Divisor of Fermat number (Euler's Result), see ProofWiki "Divisor of
Fermat Number/Euler's Result", 24-Jul-2021,
https://proofwiki.org/wiki/Divisor_of_Fermat_Number/Euler's_Result):
"Let Fn be a Fermat number. Let
m be divisor of Fn. Then m is in the
form: k*2^(n+1)+1 where k is a positive integer." Here, however, k
must
be a nonnegative integer, because k must be 0 to represent 1 (which is a
divisor of Fn ).
Historical Note: In 1747, Leonhard Paul Euler proved that a divisor of a Fermat number Fn is always in the form kx2^(n+1)+1. This was later refined to k*2^(n+2)+1 by François Édouard Anatole Lucas, see fmtnofac2 40019. (Contributed by AV, 30-Jul-2021.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1)) | ||
30-Jul-2021 | fmtnofac2 40019 | Divisor of Fermat number (Euler's Result refined by François Édouard Anatole Lucas), see fmtnofac1 40020: Let Fn be a Fermat number. Let m be divisor of Fn. Then m is in the form: k*2^(n+2)+1 where k is a nonnegative integer. (Contributed by AV, 30-Jul-2021.) |
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 2))) + 1)) | ||
30-Jul-2021 | fmtnofac2lem 40018 | Lemma for fmtnofac2 40019 (Induction step). (Contributed by AV, 30-Jul-2021.) |
⊢ ((𝑦 ∈ (ℤ≥‘2) ∧ 𝑧 ∈ (ℤ≥‘2)) → ((((𝑁 ∈ (ℤ≥‘2) ∧ 𝑦 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑦 = ((𝑘 · (2↑(𝑁 + 2))) + 1)) ∧ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑧 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑧 = ((𝑘 · (2↑(𝑁 + 2))) + 1))) → ((𝑁 ∈ (ℤ≥‘2) ∧ (𝑦 · 𝑧) ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 (𝑦 · 𝑧) = ((𝑘 · (2↑(𝑁 + 2))) + 1)))) | ||
30-Jul-2021 | fmtno5 40007 | The 5 th Fermat number. (Contributed by AV, 30-Jul-2021.) |
⊢ (FermatNo‘5) = ;;;;;;;;;4294967297 | ||
30-Jul-2021 | fmtno5lem4 40006 | Lemma 4 for fmtno5 40007. (Contributed by AV, 30-Jul-2021.) |
⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296 | ||
30-Jul-2021 | decaddm10 11454 | The sum of two multiples of 10 is a multiple of 10. (Contributed by AV, 30-Jul-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 ⇒ ⊢ (;𝐴0 + ;𝐵0) = ;(𝐴 + 𝐵)0 | ||
30-Jul-2021 | muladd11r 10128 | A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 1) · (𝐵 + 1)) = (((𝐴 · 𝐵) + (𝐴 + 𝐵)) + 1)) | ||
29-Jul-2021 | fmtnorec2 39993 | The second recurrence relation for Fermat numbers, see ProofWiki "Product of Sequence of Fermat Numbers plus 2", 29-Jul-2021, https://proofwiki.org/wiki/Product_of_Sequence_of_Fermat_Numbers_plus_2 or Wikipedia "Fermat number", 29-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 29-Jul-2021.) |
⊢ (𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2)) | ||
29-Jul-2021 | fmtnorec2lem 39992 | Lemma for fmtnorec2 39993 (induction step). (Contributed by AV, 29-Jul-2021.) |
⊢ (𝑦 ∈ ℕ0 → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2))) | ||
29-Jul-2021 | fmtnom1nn 39982 | A Fermat number minus one is a power of a power of two. (Contributed by AV, 29-Jul-2021.) |
⊢ (𝑁 ∈ ℕ0 → ((FermatNo‘𝑁) − 1) = (2↑(2↑𝑁))) | ||
28-Jul-2021 | fmtnole4prm 40028 | The first five Fermat numbers are prime. (Contributed by AV, 28-Jul-2021.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 4) → (FermatNo‘𝑁) ∈ ℙ) | ||
28-Jul-2021 | fmtnofz04prm 40027 | The first five Fermat numbers are prime, see remark in [ApostolNT] p. 7. (Contributed by AV, 28-Jul-2021.) |
⊢ (𝑁 ∈ (0...4) → (FermatNo‘𝑁) ∈ ℙ) | ||
28-Jul-2021 | 65537prm 40026 | 65537 is a prime number (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.) |
⊢ ;;;;65537 ∈ ℙ | ||
28-Jul-2021 | fmtno4prm 40025 | The 4-th Fermat number (65537) is a prime (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.) |
⊢ (FermatNo‘4) ∈ ℙ | ||
28-Jul-2021 | fmtno4prmfac193 40023 | If P was a (prime) factor of the fourth Fermat number, it would be 193. (Contributed by AV, 28-Jul-2021.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → 𝑃 = ;;193) | ||
28-Jul-2021 | fmtno4prmfac 40022 | If P was a (prime) factor of the fourth Fermat number less than the square root of the fourth Fermat number, it would be either 65 or 129 or 193. (Contributed by AV, 28-Jul-2021.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → (𝑃 = ;65 ∨ 𝑃 = ;;129 ∨ 𝑃 = ;;193)) | ||
28-Jul-2021 | fmtno4sqrt 40021 | The floor of the square root of the fourth Fermat number is 256. (Contributed by AV, 28-Jul-2021.) |
⊢ (⌊‘(√‘(FermatNo‘4))) = ;;256 | ||
28-Jul-2021 | fmtnosqrt 39989 | The floor of the square root of a Fermat number. (Contributed by AV, 28-Jul-2021.) |
⊢ (𝑁 ∈ ℕ → (⌊‘(√‘(FermatNo‘𝑁))) = (2↑(2↑(𝑁 − 1)))) | ||
28-Jul-2021 | sqrtpwpw2p 39988 | The floor of the square root of 2 to the power of 2 to the power of a positive integer plus a bounded nonnegative integer. (Contributed by AV, 28-Jul-2021.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < ((2↑((2↑(𝑁 − 1)) + 1)) + 1)) → (⌊‘(√‘((2↑(2↑𝑁)) + 𝑀))) = (2↑(2↑(𝑁 − 1)))) | ||
28-Jul-2021 | fzo1to4tp 12423 | A half-open integer range from 1 to 4 is an unordered triple. (Contributed by AV, 28-Jul-2021.) |
⊢ (1..^4) = {1, 2, 3} | ||
27-Jul-2021 | eliuni 4462 | Membership in an indexed union, one way. (Contributed by JJ, 27-Jul-2021.) |
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐶) → 𝐸 ∈ ∪ 𝑥 ∈ 𝐷 𝐵) | ||
26-Jul-2021 | fmtnoprmfac2 40017 | Divisor of Fermat number (special form of Lucas' result, see fmtnofac2 40019): Let Fn be a Fermat number. Let p be a prime divisor of Fn. Then p is in the form: k*2^(n+2)+1 where k is a positive integer. (Contributed by AV, 26-Jul-2021.) |
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 2))) + 1)) | ||
26-Jul-2021 | fmtnoprmfac2lem1 40016 | Lemma for fmtnoprmfac2 40017. (Contributed by AV, 26-Jul-2021.) |
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = 1) | ||
26-Jul-2021 | fmtnoodd 39983 | Each Fermat number is odd. (Contributed by AV, 26-Jul-2021.) |
⊢ (𝑁 ∈ ℕ0 → ¬ 2 ∥ (FermatNo‘𝑁)) | ||
26-Jul-2021 | knoppcld 31665 | Closure theorem for Knopp's function. (Contributed by Asger C. Ipsen, 26-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐶) < 1) ⇒ ⊢ (𝜑 → (𝑊‘𝐴) ∈ ℂ) | ||
26-Jul-2021 | nndivides 14828 | Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.) |
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ ℕ (𝑛 · 𝑀) = 𝑁)) | ||
26-Jul-2021 | trss 4689 | An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.) (Proof shortened by JJ, 26-Jul-2021.) |
⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | ||
26-Jul-2021 | int0 4425 | The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.) |
⊢ ∩ ∅ = V | ||
26-Jul-2021 | elintg 4418 | Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.) (Proof shortened by JJ, 26-Jul-2021.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)) | ||
26-Jul-2021 | ssuni 4395 | Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by JJ, 26-Jul-2021.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶) | ||
26-Jul-2021 | dfnfc2 4390 | An alternative statement of the effective freeness of a class 𝐴, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof shortened by JJ, 26-Jul-2021.) |
⊢ (∀𝑥 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)) | ||
25-Jul-2021 | fmtnoprmfac1 40015 | Divisor of Fermat number (special form of Euler's result, see fmtnofac1 40020): Let Fn be a Fermat number. Let p be a prime divisor of Fn. Then p is in the form: k*2^(n+1)+1 where k is a positive integer. (Contributed by AV, 25-Jul-2021.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)) | ||
25-Jul-2021 | fmtnoprmfac1lem 40014 | Lemma for fmtnoprmfac1 40015: The order of 2 modulo a prime that divides the n-th Fermat number is 2^(n+1). (Contributed by AV, 25-Jul-2021.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1))) | ||
25-Jul-2021 | odz2prm2pw 40013 | Any power of two is coprime to any prime not being two. (Contributed by AV, 25-Jul-2021.) |
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ (((2↑(2↑𝑁)) mod 𝑃) ≠ 1 ∧ ((2↑(2↑(𝑁 + 1))) mod 𝑃) = 1)) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1))) | ||
25-Jul-2021 | dvdsprmpweqle 15428 | If a positive integer divides a prime power, it is a prime power with a smaller exponent. (Contributed by AV, 25-Jul-2021.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴 ∥ (𝑃↑𝑁) → ∃𝑛 ∈ ℕ0 (𝑛 ≤ 𝑁 ∧ 𝐴 = (𝑃↑𝑛)))) | ||
25-Jul-2021 | dvdsprmpweq 15426 | If a positive integer divides a prime power, it is a prime power. (Contributed by AV, 25-Jul-2021.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴 ∥ (𝑃↑𝑁) → ∃𝑛 ∈ ℕ0 𝐴 = (𝑃↑𝑛))) | ||
25-Jul-2021 | summodnegmod 14850 | The sum of two integers modulo a positive integer equals zero iff the first of the two integers equals the negative of the other integer modulo the positive integer. (Contributed by AV, 25-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 𝐵) mod 𝑁) = 0 ↔ (𝐴 mod 𝑁) = (-𝐵 mod 𝑁))) | ||
24-Jul-2021 | fmtno4nprmfac193 40024 | 193 is not a (prime) factor of the fourth Fermat number. (Contributed by AV, 24-Jul-2021.) |
⊢ ¬ ;;193 ∥ (FermatNo‘4) | ||
24-Jul-2021 | dvdsnprmd 15241 | If a number is divisible by an integer greater than 1 and less then the number, the number is not prime. (Contributed by AV, 24-Jul-2021.) |
⊢ (𝜑 → 1 < 𝐴) & ⊢ (𝜑 → 𝐴 < 𝑁) & ⊢ (𝜑 → 𝐴 ∥ 𝑁) ⇒ ⊢ (𝜑 → ¬ 𝑁 ∈ ℙ) | ||
23-Jul-2021 | wl-ax13lem1 32466 | A version of ax-wl-13v 32465 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. This proof bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 23-Jul-2021.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | ||
23-Jul-2021 | ax-wl-13v 32465 |
A version of ax13v 2235 with a distinctor instead of a distinct
variable
expression.
Had we additonally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1827. So essentially this theorem states, that a distinct variable condition between set variables can be replaced with a distinctor expression. (Contributed by Wolf Lammen, 23-Jul-2021.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||
23-Jul-2021 | preqsn 4331 | Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶)) | ||
23-Jul-2021 | sneqrg 4310 | Closed form of sneqr 4311. (Contributed by Scott Fenton, 1-Apr-2011.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) | ||
23-Jul-2021 | eqsn 4301 | Two ways to express that a nonempty set equals a singleton. (Contributed by NM, 15-Dec-2007.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ (𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)) | ||
23-Jul-2021 | prss 4291 | A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶) | ||
23-Jul-2021 | diftpsn3 4273 | Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵}) | ||
23-Jul-2021 | prnzg 4254 | A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅) | ||
23-Jul-2021 | disjpr2 4194 | The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ (((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅) | ||
23-Jul-2021 | eqoreldif 4172 | An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ 𝐶 ↔ (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵})))) | ||
23-Jul-2021 | elpr2 4147 | A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | ||
22-Jul-2021 | fmtno5nprm 40033 | The 5 th Fermat number is a not a prime. (Contributed by AV, 22-Jul-2021.) |
⊢ (FermatNo‘5) ∉ ℙ | ||
22-Jul-2021 | fmtno5fac 40032 | The factorisation of the 5 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 22-Jul-2021.) |
⊢ (FermatNo‘5) = (;;;;;;6700417 · ;;641) | ||
22-Jul-2021 | fmtno5faclem3 40031 | Lemma 3 for fmtno5fac 40032. (Contributed by AV, 22-Jul-2021.) |
⊢ (;;;;;;;;402025020 + ;;;;;;;26801668) = ;;;;;;;;428826688 | ||
22-Jul-2021 | fmtno5faclem2 40030 | Lemma 2 for fmtno5fac 40032. (Contributed by AV, 22-Jul-2021.) |
⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502 | ||
22-Jul-2021 | fmtno5faclem1 40029 | Lemma 1 for fmtno5fac 40032. (Contributed by AV, 22-Jul-2021.) |
⊢ (;;;;;;6700417 · 4) = ;;;;;;;26801668 | ||
22-Jul-2021 | fmtno5lem3 40005 | Lemma 3 for fmtno5 40007. (Contributed by AV, 22-Jul-2021.) |
⊢ (;;;;65536 · 3) = ;;;;;196608 | ||
22-Jul-2021 | fmtno5lem2 40004 | Lemma 2 for fmtno5 40007. (Contributed by AV, 22-Jul-2021.) |
⊢ (;;;;65536 · 5) = ;;;;;327680 | ||
22-Jul-2021 | fmtno5lem1 40003 | Lemma 1 for fmtno5 40007. (Contributed by AV, 22-Jul-2021.) |
⊢ (;;;;65536 · 6) = ;;;;;393216 | ||
22-Jul-2021 | fmtnorec1 39987 | The first recurrence relation for Fermat numbers, see Wikipedia "Fermat number", https://en.wikipedia.org/wiki/Fermat_number#Basic_properties, 22-Jul-2021. (Contributed by AV, 22-Jul-2021.) |
⊢ (𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = ((((FermatNo‘𝑁) − 1)↑2) + 1)) | ||
22-Jul-2021 | decmul10addOLD 11470 | Obsolete proof of decmul10add 11469 as of 6-Sep-2021. (Contributed by AV, 22-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑀 ∈ ℕ0 & ⊢ 𝐸 = (𝑀 · 𝐴) & ⊢ 𝐹 = (𝑀 · 𝐵) ⇒ ⊢ (𝑀 · ;𝐴𝐵) = (;𝐸0 + 𝐹) | ||
22-Jul-2021 | decmul1OLD 11462 | Obsolete proof of decmul1 11461 as of 6-Sep-2021. (Contributed by AV, 22-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝑃 ∈ ℕ0 & ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 = ;𝐴𝐵 & ⊢ 𝐷 ∈ ℕ0 & ⊢ (𝐴 · 𝑃) = 𝐶 & ⊢ (𝐵 · 𝑃) = 𝐷 ⇒ ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 | ||
22-Jul-2021 | decsubiOLD 11460 | Obsolete proof of decsubi 11459 as of 6-Sep-2021. (Contributed by AV, 22-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑀 = ;𝐴𝐵 & ⊢ (𝐴 + 1) = 𝐷 & ⊢ (𝐵 − 𝑁) = 𝐶 ⇒ ⊢ (𝑀 − 𝑁) = ;𝐴𝐶 | ||
21-Jul-2021 | wilthimp 24598 | The forward implication of Wilson's theorem wilth 24597 (see wilthlem3 24596), expressed using the modulo operation: For any prime 𝑝 we have (𝑝 − 1)!≡ − 1 (mod 𝑝), see theorem 5.24 in [ApostolNT] p. 116. (Contributed by AV, 21-Jul-2021.) |
⊢ (𝑃 ∈ ℙ → ((!‘(𝑃 − 1)) mod 𝑃) = (-1 mod 𝑃)) | ||
21-Jul-2021 | mod2eq0even 14908 | An integer is 0 modulo 2 iff it is even (i.e. divisible by 2), see example 2 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.) |
⊢ (𝑁 ∈ ℤ → ((𝑁 mod 2) = 0 ↔ 2 ∥ 𝑁)) | ||
21-Jul-2021 | modmulconst 14851 | Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) ∧ 𝑀 ∈ ℕ) → ((𝐴 mod 𝑀) = (𝐵 mod 𝑀) ↔ ((𝐶 · 𝐴) mod (𝐶 · 𝑀)) = ((𝐶 · 𝐵) mod (𝐶 · 𝑀)))) | ||
21-Jul-2021 | zmod1congr 12549 | Two arbitrary integers are congruent modulo 1, see example 4 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 mod 1) = (𝐵 mod 1)) | ||
20-Jul-2021 | 2lgsoddprm 24941 | The second supplement to the law of quadratic reciprocity for odd primes (common representation, see theorem 9.5 in [ApostolNT] p. 181): The Legendre symbol for 2 at an odd prime is minus one to the power of the square of the odd prime minus one divided by eight ((2 /L 𝑃) = -1^(((P^2)-1)/8) ). (Contributed by AV, 20-Jul-2021.) |
⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 /L 𝑃) = (-1↑(((𝑃↑2) − 1) / 8))) | ||
20-Jul-2021 | 2lgsoddprmlem4 24940 | Lemma 4 for 2lgsoddprm 24941. (Contributed by AV, 20-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (2 ∥ (((𝑁↑2) − 1) / 8) ↔ (𝑁 mod 8) ∈ {1, 7})) | ||
20-Jul-2021 | 2lgsoddprmlem3 24939 | Lemma 3 for 2lgsoddprm 24941. (Contributed by AV, 20-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ 𝑅 = (𝑁 mod 8)) → (2 ∥ (((𝑅↑2) − 1) / 8) ↔ 𝑅 ∈ {1, 7})) | ||
20-Jul-2021 | 2lgsoddprmlem3d 24938 | Lemma 4 for 2lgsoddprmlem3 24939. (Contributed by AV, 20-Jul-2021.) |
⊢ (((7↑2) − 1) / 8) = (2 · 3) | ||
20-Jul-2021 | 2lgsoddprmlem3c 24937 | Lemma 3 for 2lgsoddprmlem3 24939. (Contributed by AV, 20-Jul-2021.) |
⊢ (((5↑2) − 1) / 8) = 3 | ||
20-Jul-2021 | 2lgsoddprmlem3b 24936 | Lemma 2 for 2lgsoddprmlem3 24939. (Contributed by AV, 20-Jul-2021.) |
⊢ (((3↑2) − 1) / 8) = 1 | ||
20-Jul-2021 | 2lgsoddprmlem3a 24935 | Lemma 1 for 2lgsoddprmlem3 24939. (Contributed by AV, 20-Jul-2021.) |
⊢ (((1↑2) − 1) / 8) = 0 | ||
20-Jul-2021 | lgsmulsqcoprm 24868 | The Legendre (Jacobi) symbol is preserved under multiplication with a square of an integer coprime to the second argument. Theorem 9.9(d) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) → (((𝐴↑2) · 𝐵) /L 𝑁) = (𝐵 /L 𝑁)) | ||
20-Jul-2021 | lgsmodeq 24867 | The Legendre (Jacobi) symbol is preserved under reduction mod 𝑛 when 𝑛 is odd. Theorem 9.9(c) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) → ((𝐴 mod 𝑁) = (𝐵 mod 𝑁) → (𝐴 /L 𝑁) = (𝐵 /L 𝑁))) | ||
20-Jul-2021 | lgsprme0 24864 | The Legendre symbol at any prime (even at 2) is 0 iff the prime does not divide the first argument. See definition in [ApostolNT] p. 179. (Contributed by AV, 20-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ ℙ) → ((𝐴 /L 𝑃) = 0 ↔ (𝐴 mod 𝑃) = 0)) | ||
20-Jul-2021 | lgssq 24862 | The Legendre symbol at a square is equal to 1. Together with lgsmod 24848 this implies that the Legendre symbol takes value 1 at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.) (Revised by AV, 20-Jul-2021.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴↑2) /L 𝑁) = 1) | ||
20-Jul-2021 | sq4e2t8 12824 | The square of 4 is 2 times 8. (Contributed by AV, 20-Jul-2021.) |
⊢ (4↑2) = (2 · 8) | ||
20-Jul-2021 | exp4a 631 | An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jul-2021.) |
⊢ (𝜑 → (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) | ||
20-Jul-2021 | exp4b 630 | An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) Shorten exp4a 631. (Revised by Wolf Lammen, 20-Jul-2021.) |
⊢ ((𝜑 ∧ 𝜓) → ((𝜒 ∧ 𝜃) → 𝜏)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) | ||
19-Jul-2021 | knoppndvlem16 31688 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 19-Jul-2021.) |
⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝐵 − 𝐴) = (((2 · 𝑁)↑-𝐽) / 2)) | ||
19-Jul-2021 | 2lgsoddprmlem2 24934 | Lemma 2 for 2lgsoddprm 24941. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ 𝑅 = (𝑁 mod 8)) → (2 ∥ (((𝑁↑2) − 1) / 8) ↔ 2 ∥ (((𝑅↑2) − 1) / 8))) | ||
19-Jul-2021 | 2lgsoddprmlem1 24933 | Lemma 1 for 2lgsoddprm 24941. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 = ((8 · 𝐴) + 𝐵)) → (((𝑁↑2) − 1) / 8) = (((8 · (𝐴↑2)) + (2 · (𝐴 · 𝐵))) + (((𝐵↑2) − 1) / 8))) | ||
19-Jul-2021 | sqoddm1div8z 14916 | A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (((𝑁↑2) − 1) / 8) ∈ ℤ) | ||
19-Jul-2021 | mulsucdiv2z 14915 | An integer multiplied with its successor divided by 2 yields an integer, i.e. an integer multiplied with its successor is even. (Contributed by AV, 19-Jul-2021.) |
⊢ (𝑁 ∈ ℤ → ((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ) | ||
19-Jul-2021 | dvdsaddre2b 14867 | Adding a multiple of the base does not affect divisibility. Variant of dvdsadd2b 14866 only requiring 𝐵 to be a real number (not necessarily an integer). (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℤ ∧ 𝐴 ∥ 𝐶)) → (𝐴 ∥ 𝐵 ↔ 𝐴 ∥ (𝐶 + 𝐵))) | ||
19-Jul-2021 | muldivbinom2 12909 | The square of a binomial with factor divided by a nonzero number. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((((𝐶 · 𝐴) + 𝐵)↑2) / 𝐶) = (((𝐶 · (𝐴↑2)) + (2 · (𝐴 · 𝐵))) + ((𝐵↑2) / 𝐶))) | ||
19-Jul-2021 | mulsubdivbinom2 12908 | The square of a binomial with factor minus a number divided by a nonzero number. (Contributed by AV, 19-Jul-2021.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((((𝐶 · 𝐴) + 𝐵)↑2) − 𝐷) / 𝐶) = (((𝐶 · (𝐴↑2)) + (2 · (𝐴 · 𝐵))) + (((𝐵↑2) − 𝐷) / 𝐶))) | ||
19-Jul-2021 | sqoddm1div8 12890 | A squared odd number minus 1 divided by 8 is the odd number multiplied with its successor divided by 2. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 = ((2 · 𝑁) + 1)) → (((𝑀↑2) − 1) / 8) = ((𝑁 · (𝑁 + 1)) / 2)) | ||
19-Jul-2021 | mulbinom2 12846 | The square of a binomial with factor. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐶 · 𝐴) + 𝐵)↑2) = ((((𝐶 · 𝐴)↑2) + ((2 · 𝐶) · (𝐴 · 𝐵))) + (𝐵↑2))) | ||
19-Jul-2021 | sqdivid 12791 | The square of a nonzero number divided by itself yields the number itself. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑2) / 𝐴) = 𝐴) | ||
19-Jul-2021 | nzadd 11302 | The sum of a real number not being an integer and an integer is not an integer. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝐴 ∈ (ℝ ∖ ℤ) ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ (ℝ ∖ ℤ)) | ||
19-Jul-2021 | imp4a 612 | An importation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Jul-2021.) |
⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) ⇒ ⊢ (𝜑 → (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))) | ||
19-Jul-2021 | imp4b 611 | An importation inference. (Contributed by NM, 26-Apr-1994.) Shorten imp4a 612. (Revised by Wolf Lammen, 19-Jul-2021.) |
⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ((𝜒 ∧ 𝜃) → 𝜏)) | ||
18-Jul-2021 | s1nz 13239 | A singleton word is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.) (Proof shortened by Kyle Wyonch, 18-Jul-2021.) |
⊢ 〈“𝐴”〉 ≠ ∅ | ||
18-Jul-2021 | cbvex2v 2275 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 18-Jul-2021.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) | ||
18-Jul-2021 | cbval2v 2273 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 18-Jul-2021.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) | ||
18-Jul-2021 | cbvexdva 2271 | Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 18-Jul-2021.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) | ||
18-Jul-2021 | cbvaldva 2269 | Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 18-Jul-2021.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
17-Jul-2021 | bj-notalbii 31783 | Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 3905, ballotlem2 29877, bnj1143 30115, hausdiag 21258. (Contributed by BJ, 17-Jul-2021.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓) | ||
17-Jul-2021 | cbvexv 2263 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 17-Jul-2021.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) | ||
17-Jul-2021 | cbvalv 2261 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 17-Jul-2021.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
16-Jul-2021 | 2lgs 24932 | The second supplement to the law of quadratic reciprocity (for the Legendre symbol extended to arbitrary primes as second argument). Two is a square modulo a prime 𝑃 iff 𝑃≡±1 (mod 8), see first case of theorem 9.5 in [ApostolNT] p. 181. This theorem justifies our definition of (𝑁 /L 2) (lgs2 24839) to some degree, by demanding that reciprocity extend to the case 𝑄 = 2. (Proposed by Mario Carneiro, 19-Jun-2015.) (Contributed by AV, 16-Jul-2021.) |
⊢ (𝑃 ∈ ℙ → ((2 /L 𝑃) = 1 ↔ (𝑃 mod 8) ∈ {1, 7})) | ||
16-Jul-2021 | 2lgslem3 24929 | Lemma 3 for 2lgs 24932. (Contributed by AV, 16-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → (𝑁 mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1)) | ||
16-Jul-2021 | 2lgslem3c1 24927 | Lemma 3 for 2lgslem3 24929. (Contributed by AV, 16-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 5) → (𝑁 mod 2) = 1) | ||
16-Jul-2021 | 2lgslem3b1 24926 | Lemma 2 for 2lgslem3 24929. (Contributed by AV, 16-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 3) → (𝑁 mod 2) = 1) | ||
16-Jul-2021 | 2lgslem3d 24924 | Lemma for 2lgslem3d1 24928. (Contributed by AV, 16-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝐾) + 7)) → 𝑁 = ((2 · 𝐾) + 2)) | ||
16-Jul-2021 | 2lgslem3c 24923 | Lemma for 2lgslem3c1 24927. (Contributed by AV, 16-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝐾) + 5)) → 𝑁 = ((2 · 𝐾) + 1)) | ||
16-Jul-2021 | 2lgslem3b 24922 | Lemma for 2lgslem3b1 24926. (Contributed by AV, 16-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝐾) + 3)) → 𝑁 = ((2 · 𝐾) + 1)) | ||
16-Jul-2021 | 2teven 14917 | A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 = (2 · 𝐴)) → 2 ∥ 𝐵) | ||
16-Jul-2021 | 2tp1odd 14914 | A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 = ((2 · 𝐴) + 1)) → ¬ 2 ∥ 𝐵) | ||
16-Jul-2021 | rabn0 3912 | Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999.) (Revised by BJ, 16-Jul-2021.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑) | ||
16-Jul-2021 | rabeq0 3911 | Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.) (Revised by BJ, 16-Jul-2021.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | ||
16-Jul-2021 | ralnex 2975 | Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) (Proof shortened by BJ, 16-Jul-2021.) |
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) | ||
16-Jul-2021 | raln 2974 | Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021.) |
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)) | ||
16-Jul-2021 | imnang 1758 | Quantified implication in terms of quantified negation of conjunction. (Contributed by BJ, 16-Jul-2021.) |
⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓)) | ||
16-Jul-2021 | biorfi 421 | A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 16-Jul-2021.) |
⊢ ¬ 𝜑 ⇒ ⊢ (𝜓 ↔ (𝜓 ∨ 𝜑)) | ||
15-Jul-2021 | 2lgslem3d1 24928 | Lemma 4 for 2lgslem3 24929. (Contributed by AV, 15-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 7) → (𝑁 mod 2) = 0) | ||
15-Jul-2021 | 2lgslem3a1 24925 | Lemma 1 for 2lgslem3 24929. (Contributed by AV, 15-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝑃 ∈ ℕ ∧ (𝑃 mod 8) = 1) → (𝑁 mod 2) = 0) | ||
15-Jul-2021 | mulp1mod1 12573 | The product of an integer and an integer greater than 1 increased by 1 is 1 modulo the integer greater than 1. (Contributed by AV, 15-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘2)) → (((𝑁 · 𝐴) + 1) mod 𝑁) = 1) | ||
15-Jul-2021 | neq0f 3885 | A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of neq0 3889 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by BJ, 15-Jul-2021.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | ||
15-Jul-2021 | eq0f 3884 | The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by BJ, 15-Jul-2021.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | ||
15-Jul-2021 | n0ii 3881 | If a set has elements, then it is not empty. Inference associated with n0i 3879. (Contributed by BJ, 15-Jul-2021.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ ¬ 𝐵 = ∅ | ||
15-Jul-2021 | eqvf 3177 | The universe contains every set. (Contributed by BJ, 15-Jul-2021.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) | ||
15-Jul-2021 | notnotri 125 | Inference associated with notnotr 124. (Contributed by NM, 27-Feb-2008.) (Proof shortened by Wolf Lammen, 15-Jul-2021.) |
⊢ ¬ ¬ 𝜑 ⇒ ⊢ 𝜑 | ||
14-Jul-2021 | 2lgslem3a 24921 | Lemma for 2lgslem3a1 24925. (Contributed by AV, 14-Jul-2021.) |
⊢ 𝑁 = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ⇒ ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑃 = ((8 · 𝐾) + 1)) → 𝑁 = (2 · 𝐾)) | ||
14-Jul-2021 | gausslemma2d 24899 | Gauss' Lemma (see also theorem 9.6 in [ApostolNT] p. 182) for integer 2: Let p be an odd prime. Let S={2,4,6,...,(p-1)}. Let n denote the number of elements of S whose least positive residue modulo p is greater than p/2. Then ( 2 | p ) = (-1)^n. (Contributed by AV, 14-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝐻 = ((𝑃 − 1) / 2) & ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) & ⊢ 𝑀 = (⌊‘(𝑃 / 4)) & ⊢ 𝑁 = (𝐻 − 𝑀) ⇒ ⊢ (𝜑 → (2 /L 𝑃) = (-1↑𝑁)) | ||
14-Jul-2021 | gausslemma2dlem0i 24889 | Auxiliary lemma 9 for gausslemma2d 24899. (Contributed by AV, 14-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝑀 = (⌊‘(𝑃 / 4)) & ⊢ 𝐻 = ((𝑃 − 1) / 2) & ⊢ 𝑁 = (𝐻 − 𝑀) ⇒ ⊢ (𝜑 → (((2 /L 𝑃) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /L 𝑃) = (-1↑𝑁))) | ||
14-Jul-2021 | nnoddn2prmb 15356 | A number is a prime number not equal to 2 iff it is an odd prime number. Conversion theorem for two representations of odd primes. (Contributed by AV, 14-Jul-2021.) |
⊢ (𝑁 ∈ (ℙ ∖ {2}) ↔ (𝑁 ∈ ℙ ∧ ¬ 2 ∥ 𝑁)) | ||
14-Jul-2021 | m1modge3gt1 12579 | Minus one modulo an integer greater than two is greater than one. (Contributed by AV, 14-Jul-2021.) |
⊢ (𝑀 ∈ (ℤ≥‘3) → 1 < (-1 mod 𝑀)) | ||
14-Jul-2021 | m1modnnsub1 12578 | Minus one modulo a positive integer is equal to the integer minus one. (Contributed by AV, 14-Jul-2021.) |
⊢ (𝑀 ∈ ℕ → (-1 mod 𝑀) = (𝑀 − 1)) | ||
14-Jul-2021 | modmuladdnn0 12576 | Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℕ0 𝐴 = ((𝑘 · 𝑀) + 𝐵))) | ||
14-Jul-2021 | modmuladdim 12575 | Implication of a decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵))) | ||
14-Jul-2021 | modmuladd 12574 | Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (0[,)𝑀) ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 ↔ ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵))) | ||
14-Jul-2021 | adddivflid 12481 | The floor of a sum of an integer and a fraction is equal to the integer iff the denominator of the fraction is less than the numerator. (Contributed by AV, 14-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴)) | ||
14-Jul-2021 | foco2 6287 | If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵 ∧ (𝐹 ∘ 𝐺):𝐴–onto→𝐶) → 𝐹:𝐵–onto→𝐶) | ||
14-Jul-2021 | funcnvqp 5866 | The converse quadruple of ordered pairs is a function if the second members are pairwise different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ ((((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉) ∧ (𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻) ∧ (𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐻) ∧ 𝐹 ≠ 𝐻)) → Fun ◡({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ∪ {〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉})) | ||
14-Jul-2021 | funtpg 5856 | A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → Fun {〈𝑋, 𝐴〉, 〈𝑌, 𝐵〉, 〈𝑍, 𝐶〉}) | ||
14-Jul-2021 | funprg 5854 | A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) | ||
14-Jul-2021 | elsnxp 5594 | Elementhood to a cartesian product with a singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ (𝑋 ∈ 𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑍 = 〈𝑋, 𝑦〉)) | ||
14-Jul-2021 | ralf0 4030 | The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ ¬ 𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) | ||
14-Jul-2021 | uneqdifeq 4009 | Two ways to say that 𝐴 and 𝐵 partition 𝐶 (when 𝐴 and 𝐵 don't overlap and 𝐴 is a part of 𝐶). (Contributed by FL, 17-Nov-2008.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 ↔ (𝐶 ∖ 𝐴) = 𝐵)) | ||
14-Jul-2021 | minel 3985 | A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶) | ||
14-Jul-2021 | ssdisj 3978 | Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅) | ||
14-Jul-2021 | npss0 3966 | No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ ¬ 𝐴 ⊊ ∅ | ||
14-Jul-2021 | rab0 3909 | Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ | ||
14-Jul-2021 | chvarv 2251 | Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by NM, 20-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 14-Jul-2021.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 | ||
13-Jul-2021 | gausslemma2dlem7 24898 | Lemma 7 for gausslemma2d 24899. (Contributed by AV, 13-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝐻 = ((𝑃 − 1) / 2) & ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) & ⊢ 𝑀 = (⌊‘(𝑃 / 4)) & ⊢ 𝑁 = (𝐻 − 𝑀) ⇒ ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1) | ||
13-Jul-2021 | gausslemma2dlem0c 24883 | Auxiliary lemma 3 for gausslemma2d 24899. (Contributed by AV, 13-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝐻 = ((𝑃 − 1) / 2) ⇒ ⊢ (𝜑 → ((!‘𝐻) gcd 𝑃) = 1) | ||
13-Jul-2021 | lgscl1 24845 | The value of the Legendre symbol is either -1 or 0 or 1. (Contributed by AV, 13-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ {-1, 0, 1}) | ||
13-Jul-2021 | zabsle1 24821 | {-1, 0, 1} is the set of all integers with absolute value at most 1. (Contributed by AV, 13-Jul-2021.) |
⊢ (𝑍 ∈ ℤ → (𝑍 ∈ {-1, 0, 1} ↔ (abs‘𝑍) ≤ 1)) | ||
13-Jul-2021 | cncongrprm 15275 | Corollary 2 of Cancellability of Congruences: Two products with a common factor are congruent modulo a prime number not dividing the common factor iff the other factors are congruent modulo the prime number. (Contributed by AV, 13-Jul-2021.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑃 ∈ ℙ ∧ ¬ 𝑃 ∥ 𝐶)) → (((𝐴 · 𝐶) mod 𝑃) = ((𝐵 · 𝐶) mod 𝑃) ↔ (𝐴 mod 𝑃) = (𝐵 mod 𝑃))) | ||
13-Jul-2021 | prmndvdsfaclt 15273 | A prime number does not divide the factorial of a nonnegative integer less than the prime number. (Contributed by AV, 13-Jul-2021.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝑁 < 𝑃 → ¬ 𝑃 ∥ (!‘𝑁))) | ||
13-Jul-2021 | cncongrcoprm 15222 | Corollary 1 of Cancellability of Congruences: Two products with a common factor are congruent modulo an integer being coprime to the common factor iff the other factors are congruent modulo the integer. (Contributed by AV, 13-Jul-2021.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ (𝐶 gcd 𝑁) = 1)) → (((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁) ↔ (𝐴 mod 𝑁) = (𝐵 mod 𝑁))) | ||
13-Jul-2021 | cncongr 15221 | Cancellability of Congruences (see ProofWiki "Cancellability of Congruences, https://proofwiki.org/wiki/Cancellability_of_Congruences, 10-Jul-2021): Two products with a common factor are congruent modulo a positive integer iff the other factors are congruent modulo the integer divided by the greates common divisor of the integer and the common factor. See also Theorem 5.4 "Cancellation law" in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ 𝑀 = (𝑁 / (𝐶 gcd 𝑁)))) → (((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁) ↔ (𝐴 mod 𝑀) = (𝐵 mod 𝑀))) | ||
13-Jul-2021 | cncongr1 15219 | One direction of the bicondition in cncongr 15221. Theorem 5.4 in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ 𝑀 = (𝑁 / (𝐶 gcd 𝑁)))) → (((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁) → (𝐴 mod 𝑀) = (𝐵 mod 𝑀))) | ||
13-Jul-2021 | undif3 3847 | An equality involving class union and class difference. The first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 17-Apr-2012.) (Proof shortened by JJ, 13-Jul-2021.) |
⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) | ||
12-Jul-2021 | elsetrecs 42244 | A set 𝐴 is an element of setrecs(𝐹) iff 𝐴 is generated by some subset of setrecs(𝐹). The proof requires both setrec1 42237 and setrec2 42241, but this theorem is not strong enough to uniquely determine setrecs(𝐹). If 𝐹 respects the subset relation, the theorem still holds if both occurrences of ∈ are replaced by ⊆ for a stronger version of the theorem. (Contributed by Emmett Weisz, 12-Jul-2021.) |
⊢ 𝐵 = setrecs(𝐹) ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) | ||
12-Jul-2021 | divgcdcoprmex 15218 | Integers divided by gcd are coprime (see ProofWiki "Integers Divided by GCD are Coprime", 11-Jul-2021, https://proofwiki.org/wiki/Integers_Divided_by_GCD_are_Coprime): Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their gcd. (Contributed by AV, 12-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ 𝑀 = (𝐴 gcd 𝐵)) → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ (𝐴 = (𝑀 · 𝑎) ∧ 𝐵 = (𝑀 · 𝑏) ∧ (𝑎 gcd 𝑏) = 1)) | ||
12-Jul-2021 | divgcdcoprm0 15217 | Integers divided by gcd are coprime. (Contributed by AV, 12-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1) | ||
11-Jul-2021 | elsetrecslem 42243 | Lemma for elsetrecs 42244. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 42242. To see why this lemma also requires setrec1 42237, consider what would happen if we replaced 𝐵 with {𝐴}. The antecedent would still hold, but the consequent would fail in general. Consider dispensing with the deduction form. (Contributed by Emmett Weisz, 11-Jul-2021.) (New usage is discouraged.) |
⊢ 𝐵 = setrecs(𝐹) ⇒ ⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥))) | ||
11-Jul-2021 | ntrneik4 37419 | Idempotence of the interior function is equivalent to stating a set, 𝑠, is a neighborhood of a point, 𝑥 is equivalent to there existing a special neighborhood, 𝑢, of 𝑥 such that a point is an element of the special neighborhood if and only if 𝑠 is also a neighborhood of the point. (Contributed by RP, 11-Jul-2021.) |
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦))))) | ||
11-Jul-2021 | ntrneik4w 37418 | Idempotence of the interior function is equivalent to saying a set is a neighborhood of a point if and only if the interior of the set is a neighborhood of a point. (Contributed by RP, 11-Jul-2021.) |
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥)))) | ||
11-Jul-2021 | ntrneiel2 37404 | Membership in iterated interior of a set is equivalent to there existing a particular neighborhood of that member such that points are members of that neighborhood if and only if the set is a neighborhood of each of those points. (Contributed by RP, 11-Jul-2021.) |
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ (𝜑 → 𝐼𝐹𝑁) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐼‘𝑆)) ↔ ∃𝑢 ∈ (𝑁‘𝑋)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦)))) | ||
11-Jul-2021 | cncongr2 15220 | The other direction of the bicondition in cncongr 15221. (Contributed by AV, 11-Jul-2021.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (𝑁 ∈ ℕ ∧ 𝑀 = (𝑁 / (𝐶 gcd 𝑁)))) → ((𝐴 mod 𝑀) = (𝐵 mod 𝑀) → ((𝐴 · 𝐶) mod 𝑁) = ((𝐵 · 𝐶) mod 𝑁))) | ||
11-Jul-2021 | congr 15216 | Definition of congruence by integer multiple (see ProofWiki "Congruence (Number Theory)", 11-Jul-2021, https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory)): An integer 𝐴 is congruent to an integer 𝐵 modulo 𝑀 if their difference is a multiple of 𝑀. See also the definition in [ApostolNT] p. 104: "... 𝑎 is congruent to 𝑏 modulo 𝑚, and we write 𝑎≡𝑏 (mod 𝑚) if 𝑚 divides the difference 𝑎 − 𝑏", or Wikipedia "Modular arithmetic - Congruence", https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence, 11-Jul-2021,: "Given an integer n > 1, called a modulus, two integers are said to be congruent modulo n, if n is a divisor of their difference (i.e., if there is an integer k such that a-b = kn)". (Contributed by AV, 11-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℕ) → ((𝐴 mod 𝑀) = (𝐵 mod 𝑀) ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = (𝐴 − 𝐵))) | ||
11-Jul-2021 | divgcdz 15071 | An integer divided by the gcd of it and a nonzero integer is an integer. (Contributed by AV, 11-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) → (𝐴 / (𝐴 gcd 𝐵)) ∈ ℤ) | ||
11-Jul-2021 | zeqzmulgcd 15070 | An integer is the product of an integer and the gcd of it and another integer. (Contributed by AV, 11-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ∃𝑛 ∈ ℤ 𝐴 = (𝑛 · (𝐴 gcd 𝐵))) | ||
10-Jul-2021 | ntrclsk4 37390 | Idempotence of the interior function is equivalent to idempotence of the closure function. (Contributed by RP, 10-Jul-2021.) |
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) ⇒ ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠))) | ||
10-Jul-2021 | coprmdvds 15204 | Euclid's Lemma (see ProofWiki "Euclid's Lemma", 10-Jul-2021, https://proofwiki.org/wiki/Euclid's_Lemma): If an integer divides the product of two integers and is coprime to one of them, then it divides the other. See also theorem 1.5 in [ApostolNT] p. 16. Generalization of euclemma 15263. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.) |
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ (𝑀 · 𝑁) ∧ (𝐾 gcd 𝑀) = 1) → 𝐾 ∥ 𝑁)) | ||
10-Jul-2021 | divgcdnnr 15075 | A positive integer divided by the gcd of it and another integer is a positive integer. (Contributed by AV, 10-Jul-2021.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ) → (𝐴 / (𝐵 gcd 𝐴)) ∈ ℕ) | ||
10-Jul-2021 | divgcdnn 15074 | A positive integer divided by the gcd of it and another integer is a positive integer. (Contributed by AV, 10-Jul-2021.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ) → (𝐴 / (𝐴 gcd 𝐵)) ∈ ℕ) | ||
10-Jul-2021 | gcd2n0cl 15069 | Closure of the gcd operator if the second operand is not 0. (Contributed by AV, 10-Jul-2021.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀 gcd 𝑁) ∈ ℕ) | ||
10-Jul-2021 | divmulasscom 10588 | An associative/commutative law for division and multiplication. (Contributed by AV, 10-Jul-2021.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = (𝐵 · ((𝐴 · 𝐶) / 𝐷))) | ||
10-Jul-2021 | divmulass 10587 | An associative law for division and multiplication. (Contributed by AV, 10-Jul-2021.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = ((𝐴 · 𝐵) · (𝐶 / 𝐷))) | ||
10-Jul-2021 | 19.42-1 2091 | One direction of 19.42 2092. (Contributed by Wolf Lammen, 10-Jul-2021.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) | ||
9-Jul-2021 | gausslemma2dlem5 24896 | Lemma 5 for gausslemma2d 24899. (Contributed by AV, 9-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝐻 = ((𝑃 − 1) / 2) & ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) & ⊢ 𝑀 = (⌊‘(𝑃 / 4)) & ⊢ 𝑁 = (𝐻 − 𝑀) ⇒ ⊢ (𝜑 → (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) mod 𝑃) = (((-1↑𝑁) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑘 · 2)) mod 𝑃)) | ||
9-Jul-2021 | gausslemma2dlem0h 24888 | Auxiliary lemma 8 for gausslemma2d 24899. (Contributed by AV, 9-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝑀 = (⌊‘(𝑃 / 4)) & ⊢ 𝐻 = ((𝑃 − 1) / 2) & ⊢ 𝑁 = (𝐻 − 𝑀) ⇒ ⊢ (𝜑 → 𝑁 ∈ ℕ0) | ||
9-Jul-2021 | gausslemma2dlem0g 24887 | Auxiliary lemma 7 for gausslemma2d 24899. (Contributed by AV, 9-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝑀 = (⌊‘(𝑃 / 4)) & ⊢ 𝐻 = ((𝑃 − 1) / 2) ⇒ ⊢ (𝜑 → 𝑀 ≤ 𝐻) | ||
9-Jul-2021 | gausslemma2dlem0f 24886 | Auxiliary lemma 6 for gausslemma2d 24899. (Contributed by AV, 9-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝑀 = (⌊‘(𝑃 / 4)) & ⊢ 𝐻 = ((𝑃 − 1) / 2) ⇒ ⊢ (𝜑 → (𝑀 + 1) ≤ 𝐻) | ||
9-Jul-2021 | gausslemma2dlem0e 24885 | Auxiliary lemma 5 for gausslemma2d 24899. (Contributed by AV, 9-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝑀 = (⌊‘(𝑃 / 4)) ⇒ ⊢ (𝜑 → (𝑀 · 2) < (𝑃 / 2)) | ||
9-Jul-2021 | gausslemma2dlem0d 24884 | Auxiliary lemma 4 for gausslemma2d 24899. (Contributed by AV, 9-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝑀 = (⌊‘(𝑃 / 4)) ⇒ ⊢ (𝜑 → 𝑀 ∈ ℕ0) | ||
9-Jul-2021 | gausslemma2dlem0b 24882 | Auxiliary lemma 2 for gausslemma2d 24899. (Contributed by AV, 9-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝐻 = ((𝑃 − 1) / 2) ⇒ ⊢ (𝜑 → 𝐻 ∈ ℕ) | ||
9-Jul-2021 | gausslemma2dlem0a 24881 | Auxiliary lemma 1 for gausslemma2d 24899. (Contributed by AV, 9-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) ⇒ ⊢ (𝜑 → 𝑃 ∈ ℕ) | ||
9-Jul-2021 | modmulmodr 12598 | The product of an integer and a real number modulo a positive real number equals the product of the integer and the real number modulo the positive real number. (Contributed by Alexander van der Vekens, 9-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → ((𝐴 · (𝐵 mod 𝑀)) mod 𝑀) = ((𝐴 · 𝐵) mod 𝑀)) | ||
9-Jul-2021 | fldiv4lem1div2 12500 | The floor of a positive integer divided by 4 is less than or equal to the half of the integer minus 1. (Contributed by AV, 9-Jul-2021.) |
⊢ (𝑁 ∈ ℕ → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) | ||
8-Jul-2021 | df3o3 37343 | Ordinal 3 , fully expanded. (Contributed by RP, 8-Jul-2021.) |
⊢ 3𝑜 = {∅, {∅}, {∅, {∅}}} | ||
8-Jul-2021 | df3o2 37342 | Ordinal 3 is the triplet containing ordinals 0, 1 and 2. (Contributed by RP, 8-Jul-2021.) |
⊢ 3𝑜 = {∅, 1𝑜, 2𝑜} | ||
8-Jul-2021 | equcomi1 33203 | Proof of equcomi 1931 from equid1 33202, avoiding use of ax-5 1827 (the only use of ax-5 1827 is via ax7 1930, so using ax-7 1922 instead would remove dependency on ax-5 1827). (Contributed by BJ, 8-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) | ||
8-Jul-2021 | gausslemma2dlem5a 24895 | Lemma for gausslemma2dlem5 24896. (Contributed by AV, 8-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝐻 = ((𝑃 − 1) / 2) & ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) & ⊢ 𝑀 = (⌊‘(𝑃 / 4)) ⇒ ⊢ (𝜑 → (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) mod 𝑃) = (∏𝑘 ∈ ((𝑀 + 1)...𝐻)(-1 · (𝑘 · 2)) mod 𝑃)) | ||
8-Jul-2021 | fldiv4p1lem1div2 12498 | The floor of an integer equal to 3 or greater than 4, increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.) |
⊢ ((𝑁 = 3 ∨ 𝑁 ∈ (ℤ≥‘5)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2)) | ||
8-Jul-2021 | divfl0 12487 | The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (⌊‘(𝐴 / 𝐵)) = 0)) | ||
8-Jul-2021 | div4p1lem1div2 11164 | An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.) |
⊢ ((𝑁 ∈ ℝ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2)) | ||
7-Jul-2021 | ssdifsn 42228 | Subset of a set with an element removed. (Contributed by Emmett Weisz, 7-Jul-2021.) |
⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴)) | ||
7-Jul-2021 | bj-genan 31778 | Generalization rule on a conjunction. Forward inference associated with 19.26 1786. (Contributed by BJ, 7-Jul-2021.) |
⊢ (𝜑 ∧ 𝜓) ⇒ ⊢ (∀𝑥𝜑 ∧ ∀𝑥𝜓) | ||
7-Jul-2021 | bj-genl 31777 | Generalization rule on the left conjunct. See 19.27 2082. (Contributed by BJ, 7-Jul-2021.) |
⊢ (𝜑 ∧ 𝜓) ⇒ ⊢ (∀𝑥𝜑 ∧ 𝜓) | ||
7-Jul-2021 | bj-genr 31776 | Generalization rule on the right conjunct. See 19.28 2083. (Contributed by BJ, 7-Jul-2021.) |
⊢ (𝜑 ∧ 𝜓) ⇒ ⊢ (𝜑 ∧ ∀𝑥𝜓) | ||
7-Jul-2021 | knoppndvlem14 31686 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 7-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖))) ≤ ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) | ||
7-Jul-2021 | fprodmodd 14567 | If all factors of two finite products are equal modulo 𝑀, the products are equal modulo 𝑀. (Contributed by AV, 7-Jul-2021.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 mod 𝑀) = (𝐶 mod 𝑀)) ⇒ ⊢ (𝜑 → (∏𝑘 ∈ 𝐴 𝐵 mod 𝑀) = (∏𝑘 ∈ 𝐴 𝐶 mod 𝑀)) | ||
7-Jul-2021 | sbcrext 3478 | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) |
⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) | ||
7-Jul-2021 | sbcimdv 3465 | Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1729). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) | ||
7-Jul-2021 | eqsbc3r 3459 | eqsbc3 3442 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴)) | ||
7-Jul-2021 | exanOLD 1776 | Obsolete proof of exan 1775 as of 8-Oct-2021. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) Reduce axiom dependencies. (Revised by BJ, 7-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑥𝜑 ∧ 𝜓) ⇒ ⊢ ∃𝑥(𝜑 ∧ 𝜓) | ||
6-Jul-2021 | young2d 42360 | Young's inequality for 𝑛 = 2, a direct application of amgmw2d 42359. (Contributed by Kunhao Zheng, 6-Jul-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝑃 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝑄 ∈ ℝ+) & ⊢ (𝜑 → ((1 / 𝑃) + (1 / 𝑄)) = 1) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ≤ (((𝐴↑𝑐𝑃) / 𝑃) + ((𝐵↑𝑐𝑄) / 𝑄))) | ||
6-Jul-2021 | clsk1indlem1 37363 | The ansatz closure function (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) does not have the K1 property of isotony. (Contributed by RP, 6-Jul-2021.) |
⊢ 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) ⇒ ⊢ ∃𝑠 ∈ 𝒫 3𝑜∃𝑡 ∈ 𝒫 3𝑜(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) | ||
6-Jul-2021 | clsk1indlem4 37362 | The ansatz closure function (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) has the K4 property of idempotence. (Contributed by RP, 6-Jul-2021.) |
⊢ 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) ⇒ ⊢ ∀𝑠 ∈ 𝒫 3𝑜(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠) | ||
6-Jul-2021 | clsk1indlem3 37361 | The ansatz closure function (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) has the K3 property of being sub-linear. (Contributed by RP, 6-Jul-2021.) |
⊢ 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) ⇒ ⊢ ∀𝑠 ∈ 𝒫 3𝑜∀𝑡 ∈ 𝒫 3𝑜(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) | ||
6-Jul-2021 | clsk1indlem2 37360 | The ansatz closure function (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) has the K2 property of expanding. (Contributed by RP, 6-Jul-2021.) |
⊢ 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) ⇒ ⊢ ∀𝑠 ∈ 𝒫 3𝑜𝑠 ⊆ (𝐾‘𝑠) | ||
6-Jul-2021 | clsk1indlem0 37359 | The ansatz closure function (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) has the K0 property of preserving the nullary union. (Contributed by RP, 6-Jul-2021.) |
⊢ 𝐾 = (𝑟 ∈ 𝒫 3𝑜 ↦ if(𝑟 = {∅}, {∅, 1𝑜}, 𝑟)) ⇒ ⊢ (𝐾‘∅) = ∅ | ||
6-Jul-2021 | bj-ax12v3ALT 31863 | Alternate proof of bj-ax12v3 31862. Uses axc11r 2175 and axc15 2291 instead of ax-12 2034. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
6-Jul-2021 | bj-ax12v3 31862 | A weak version of ax-12 2034 which is stronger than ax12v 2035. Note that if one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 1926), then bj-ax12v3 31862 implies ax-5 1827 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 31863. (Contributed by BJ, 6-Jul-2021.) |
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
6-Jul-2021 | bj-axc16g16 31861 | Proof of axc16g 2119 from { ax-1 6-- ax-7 1922, axc16 2120 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | ||
6-Jul-2021 | axc11n11r 31860 |
Proof of axc11n 2295 from { ax-1 6--
ax-7 1922, axc9 2290, axc11r 2175 } (note
that axc16 2120 is provable from { ax-1 6--
ax-7 1922, axc11r 2175 }).
Note that axc11n 2295 proves (over minimal calculus) that axc11 2302 and axc11r 2175 are equivalent. Therefore, axc11n11 31859 and axc11n11r 31860 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2302 appears slightly stronger since axc11n11r 31860 requires axc9 2290 while axc11n11 31859 does not). (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
6-Jul-2021 | axc11n11 31859 | Proof of axc11n 2295 from { ax-1 6-- ax-7 1922, axc11 2302 } . Almost identical to axc11nfromc11 33229. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
6-Jul-2021 | knoppndvlem15 31687 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 6-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ (abs‘((𝑊‘𝐵) − (𝑊‘𝐴)))) | ||
6-Jul-2021 | axc11vOLD 2126 | Obsolete proof of axc11v 2123 as of 11-Oct-2021. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
6-Jul-2021 | axc11rvOLD 2125 | Obsolete proof of axc11rv 2124 as of 11-Oct-2011. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑)) | ||
5-Jul-2021 | clsk1independent 37364 | For generalized closure functions, property K1 (isotony) is independent of the properties K0, K2, K3, K4. This contradicts a claim which appears in preprints of Table 2 in Bärbel M. R. Stadler and Peter F. Stadler. "Generalized Topological Spaces in Evolutionary Theory and Combinatorial Chemistry." J. Chem. Inf. Comput. Sci., 42:577-585, 2002. Proceedings MCC 2001, Dubrovnik. The same table row implying K1 follows from the other four appears in the supplemental materials Bärbel M. R. Stadler and Peter F. Stadler. "Basic Properties of Closure Spaces" 2001 on page 12. (Contributed by RP, 5-Jul-2021.) |
⊢ (𝜑 ↔ (𝑘‘∅) = ∅) & ⊢ (𝜓 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡))) & ⊢ (𝜒 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘‘𝑠)) & ⊢ (𝜃 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡))) & ⊢ (𝜏 ↔ ∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠)) ⇒ ⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓) | ||
5-Jul-2021 | uneqsn 37341 | If a union of classes is equal to a singleton then at least one class is equal to the singleton while the other may be equal to the empty set. (Contributed by RP, 5-Jul-2021.) |
⊢ ((𝐴 ∪ 𝐵) = {𝐶} ↔ ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶}))) | ||
5-Jul-2021 | andi3or 37340 | Distribute over triple disjunction. (Contributed by RP, 5-Jul-2021.) |
⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒 ∨ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃))) | ||
5-Jul-2021 | or3or 37339 | Decompose disjunction into three cases. (Contributed by RP, 5-Jul-2021.) |
⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓))) | ||
5-Jul-2021 | knoppndvlem13 31685 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → 𝐶 ≠ 0) | ||
5-Jul-2021 | knoppndvlem12 31684 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 29-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) | ||
5-Jul-2021 | knoppndvlem11 31683 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 28-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖))) ≤ ((abs‘(𝐵 − 𝐴)) · Σ𝑖 ∈ (0...(𝐽 − 1))(((2 · 𝑁) · (abs‘𝐶))↑𝑖))) | ||
5-Jul-2021 | knoppndvlem10 31682 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2)) | ||
5-Jul-2021 | knoppndvlem9 31681 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → ¬ 2 ∥ 𝑀) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = ((𝐶↑𝐽) / 2)) | ||
5-Jul-2021 | knoppndvlem8 31680 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 2 ∥ 𝑀) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = 0) | ||
5-Jul-2021 | knoppndvlem7 31679 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = ((𝐶↑𝐽) · (𝑇‘(𝑀 / 2)))) | ||
5-Jul-2021 | knoppndvlem6 31678 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝑊‘𝐴) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖)) | ||
5-Jul-2021 | knoppndvlem5 31677 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) ∈ ℝ) | ||
5-Jul-2021 | knoppndvlem4 31676 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → seq0( + , (𝐹‘𝐴)) ⇝ (𝑊‘𝐴)) | ||
5-Jul-2021 | knoppndvlem2 31674 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℤ) & ⊢ (𝜑 → 𝐽 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐽 < 𝐼) ⇒ ⊢ (𝜑 → (((2 · 𝑁)↑𝐼) · ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)) ∈ ℤ) | ||
5-Jul-2021 | knoppndvlem1 31673 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐽 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ) | ||
5-Jul-2021 | knoppcn 31664 | The continuous nowhere differentiable function 𝑊 ( Knopp, K. (1918). Math. Z. 2, 1-26 ) is, in fact, continuous. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐶) < 1) ⇒ ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℂ)) | ||
5-Jul-2021 | knoppcnlem11 31663 | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))):ℕ0⟶(ℝ–cn→ℂ)) | ||
5-Jul-2021 | knoppcnlem10 31662 | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑀)) ∈ ((topGen‘ran (,)) Cn (TopOpen‘ℂfld))) | ||
5-Jul-2021 | knoppcnlem9 31661 | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐶) < 1) ⇒ ⊢ (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))(⇝𝑢‘ℝ)𝑊) | ||
5-Jul-2021 | knoppcnlem8 31660 | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))):ℕ0⟶(ℂ ↑𝑚 ℝ)) | ||
5-Jul-2021 | knoppcnlem7 31659 | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → (seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))‘𝑀) = (𝑤 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑤))‘𝑀))) | ||
5-Jul-2021 | knoppcnlem6 31658 | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐶) < 1) ⇒ ⊢ (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))) ∈ dom (⇝𝑢‘ℝ)) | ||
5-Jul-2021 | knoppcnlem5 31657 | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))):ℕ0⟶(ℂ ↑𝑚 ℝ)) | ||
5-Jul-2021 | knoppcnlem4 31656 | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → (abs‘((𝐹‘𝐴)‘𝑀)) ≤ ((𝑚 ∈ ℕ0 ↦ ((abs‘𝐶)↑𝑚))‘𝑀)) | ||
5-Jul-2021 | knoppcnlem3 31655 | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴)‘𝑀) ∈ ℝ) | ||
5-Jul-2021 | knoppcnlem2 31654 | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))) ∈ ℝ) | ||
5-Jul-2021 | gausslemma2dlem1 24891 | Lemma 1 for gausslemma2d 24899. (Contributed by AV, 5-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝐻 = ((𝑃 − 1) / 2) & ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) ⇒ ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) | ||
5-Jul-2021 | prm23ge5 15358 | A prime is either 2 or 3 or greater than or equal to 5. (Contributed by AV, 5-Jul-2021.) |
⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ≥‘5))) | ||
5-Jul-2021 | prm23lt5 15357 | A prime less than 5 is either 2 or 3. (Contributed by AV, 5-Jul-2021.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑃 < 5) → (𝑃 = 2 ∨ 𝑃 = 3)) | ||
5-Jul-2021 | fldiv4lem1div2uz2 12499 | The floor of an integer greater than 1, divided by 4 is less than or equal to the half of the integer minus 1. (Contributed by AV, 5-Jul-2021.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → (⌊‘(𝑁 / 4)) ≤ ((𝑁 − 1) / 2)) | ||
5-Jul-2021 | rneqdmfinf1o 8127 | Any function from a finite set onto the same set must be a bijection. (Contributed by AV, 5-Jul-2021.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴–1-1-onto→𝐴) | ||
4-Jul-2021 | gausslemma2dlem3 24893 | Lemma 3 for gausslemma2d 24899. (Contributed by AV, 4-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝐻 = ((𝑃 − 1) / 2) & ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) & ⊢ 𝑀 = (⌊‘(𝑃 / 4)) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) = (𝑃 − (𝑘 · 2))) | ||
4-Jul-2021 | gausslemma2dlem2 24892 | Lemma 2 for gausslemma2d 24899. (Contributed by AV, 4-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝐻 = ((𝑃 − 1) / 2) & ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) & ⊢ 𝑀 = (⌊‘(𝑃 / 4)) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = (𝑘 · 2)) | ||
4-Jul-2021 | flodddiv4t2lthalf 14978 | The floor of an odd number divided by 4, multiplied by 2 is less than the half of the odd number. (Contributed by AV, 4-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → ((⌊‘(𝑁 / 4)) · 2) < (𝑁 / 2)) | ||
4-Jul-2021 | flodddiv4lt 14977 | The floor of an odd number divided by 4 is less than the odd number divided by 4. (Contributed by AV, 4-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (⌊‘(𝑁 / 4)) < (𝑁 / 4)) | ||
4-Jul-2021 | fldivndvdslt 14976 | The floor of an integer divided by a nonzero integer not dividing the first integer is less than the integer divided by the positive integer. (Contributed by AV, 4-Jul-2021.) |
⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℤ ∧ 𝐿 ≠ 0) ∧ ¬ 𝐿 ∥ 𝐾) → (⌊‘(𝐾 / 𝐿)) < (𝐾 / 𝐿)) | ||
4-Jul-2021 | 4dvdseven 14947 | An integer which is divisible by 4 is an even integer. (Contributed by AV, 4-Jul-2021.) |
⊢ (4 ∥ 𝑁 → 2 ∥ 𝑁) | ||
4-Jul-2021 | z4even 14946 | 4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.) |
⊢ 2 ∥ 4 | ||
4-Jul-2021 | ax12 2292 | Rederivation of axiom ax-12 2034 from ax12v 2035, axc11r 2175, and other axioms. (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.) (Proof shortened by BJ, 4-Jul-2021.) |
⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
4-Jul-2021 | stdpc5OLDOLD 2205 | Obsolete proof of stdpc5 2063 as of 6-Oct-2021. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 4-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) | ||
4-Jul-2021 | 19.21t-1OLD 2200 | One direction of the bi-conditional in 19.21t 2061. Unlike the reverse implication, it does not depend on ax-10 2006. Obsolete as of 6-Oct-2021 (Contributed by Wolf Lammen, 4-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | ||
4-Jul-2021 | stdpc5OLD 2064 | Obsolete proof of stdpc5 2063 as of 11-Oct-2021. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 4-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) | ||
2-Jul-2021 | axc11n 2295 | Derive set.mm's original ax-c11n 33191 from others. Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). If a disjoint variable condition is added on 𝑥 and 𝑦, then this becomes an instance of aevlem 1968. Use aecom 2299 instead when this does not lengthen the proof. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Revised by Wolf Lammen, 30-Nov-2019.) (Proof shortened by BJ, 29-Mar-2021.) (Proof shortened by Wolf Lammen, 2-Jul-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
1-Jul-2021 | gausslemma2dlem1a 24890 | Lemma for gausslemma2dlem1 24891. (Contributed by AV, 1-Jul-2021.) |
⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝐻 = ((𝑃 − 1) / 2) & ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2)))) ⇒ ⊢ (𝜑 → ran 𝑅 = (1...𝐻)) | ||
1-Jul-2021 | halfleoddlt 14924 | An integer is greater than half of an odd number iff it is greater than or equal to the half of the odd number. (Contributed by AV, 1-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ 𝑀 ∈ ℤ) → ((𝑁 / 2) ≤ 𝑀 ↔ (𝑁 / 2) < 𝑀)) | ||
1-Jul-2021 | zltaddlt1le 12195 | The sum of an integer and a real number between 0 and 1 is less than or equal to a second integer iff the sum is less than the second integer. (Contributed by AV, 1-Jul-2021.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ (0(,)1)) → ((𝑀 + 𝐴) < 𝑁 ↔ (𝑀 + 𝐴) ≤ 𝑁)) | ||
1-Jul-2021 | addlelt 11818 | If the sum of a real number and a positive real number is less than or equal to a third real number, the first real number is less than the third real number. (Contributed by AV, 1-Jul-2021.) |
⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → ((𝑀 + 𝐴) ≤ 𝑁 → 𝑀 < 𝑁)) | ||
1-Jul-2021 | muldivdir 10599 | Distribution of division over addition with a multiplication. (Contributed by AV, 1-Jul-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((𝐶 · 𝐴) + 𝐵) / 𝐶) = (𝐴 + (𝐵 / 𝐶))) | ||
29-Jun-2021 | ltoddhalfle 14923 | An integer is less than half of an odd number iff it is less than or equal to the half of the predecessor of the odd number (which is an even number). (Contributed by AV, 29-Jun-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁 ∧ 𝑀 ∈ ℤ) → (𝑀 < (𝑁 / 2) ↔ 𝑀 ≤ ((𝑁 − 1) / 2))) | ||
29-Jun-2021 | ledivge1le 11777 | If a number is less than or equal to another number, the number divided by a positive number greater than or equal to one is less than or equal to the other number. (Contributed by AV, 29-Jun-2021.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ (𝐶 ∈ ℝ+ ∧ 1 ≤ 𝐶)) → (𝐴 ≤ 𝐵 → (𝐴 / 𝐶) ≤ 𝐵)) | ||
29-Jun-2021 | divle1le 11776 | A real number divided by a positive real number is less than or equal to 1 iff the real number is less than or equal to the positive real number. (Contributed by AV, 29-Jun-2021.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) ≤ 1 ↔ 𝐴 ≤ 𝐵)) | ||
28-Jun-2021 | neik0imk0p 37354 | Kuratowski's K0 axiom implies K0'. Neighborhood version. Also a proof the dual KA axiom imples KA' when considering the convergents. (Contributed by RP, 28-Jun-2021.) |
⊢ (∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅) | ||
28-Jun-2021 | oddn2prm 15355 | A prime not equal to 2 is odd. (Contributed by AV, 28-Jun-2021.) |
⊢ (𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑁) | ||
28-Jun-2021 | nnoddn2prm 15354 | A prime not equal to 2 is an odd positive integer. (Contributed by AV, 28-Jun-2021.) |
⊢ (𝑁 ∈ (ℙ ∖ {2}) → (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) | ||
28-Jun-2021 | nnoddm1d2 14940 | A positive integer is odd iff its successor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.) |
⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℕ)) | ||
28-Jun-2021 | nn0oddm1d2 14939 | A positive integer is odd iff its predecessor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.) |
⊢ (𝑁 ∈ ℕ0 → (¬ 2 ∥ 𝑁 ↔ ((𝑁 − 1) / 2) ∈ ℕ0)) | ||
28-Jun-2021 | nnehalf 14934 | The half of an even positive integer is a positive integer. (Contributed by AV, 28-Jun-2021.) |
⊢ ((𝑁 ∈ ℕ ∧ 2 ∥ 𝑁) → (𝑁 / 2) ∈ ℕ) | ||
28-Jun-2021 | nn0ehalf 14933 | The half of an even nonnegative integer is a nonnegative integer. (Contributed by AV, 22-Jun-2020.) (Revised by AV, 28-Jun-2021.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 2 ∥ 𝑁) → (𝑁 / 2) ∈ ℕ0) | ||
28-Jun-2021 | subhalfhalf 11143 | Subtracting the half of a number from the number yields the half of the number. (Contributed by AV, 28-Jun-2021.) |
⊢ (𝐴 ∈ ℂ → (𝐴 − (𝐴 / 2)) = (𝐴 / 2)) | ||
27-Jun-2021 | neicvgfv 37439 | The value of the neighborhoods (convergents) in terms of the the convergents (neighborhoods) function. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) & ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) & ⊢ (𝜑 → 𝑁𝐻𝑀) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)}) | ||
27-Jun-2021 | neicvgnex 37436 | If the neighborhoods and convergents functions are related, the neighborhoods function exists. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) & ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) & ⊢ (𝜑 → 𝑁𝐻𝑀) ⇒ ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑𝑚 𝐵)) | ||
27-Jun-2021 | neicvgmex 37435 | If the neighborhoods and convergents functions are related, the convergents function exists. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵) & ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) & ⊢ (𝜑 → 𝑁𝐻𝑀) ⇒ ⊢ (𝜑 → 𝑀 ∈ (𝒫 𝒫 𝐵 ↑𝑚 𝐵)) | ||
27-Jun-2021 | clsneifv4 37429 | Value of the the closure (interior) function in terms of the neighborhoods (convergents) function. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → (𝐾‘𝑆) = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)}) | ||
27-Jun-2021 | clsneifv3 37428 | Value of the neighborhoods (convergents) in terms of the closure (interior) function. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑠))}) | ||
27-Jun-2021 | clsneinex 37425 | If closure and neighborhoods functions are related, the neighborhoods function exists. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) ⇒ ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑𝑚 𝐵)) | ||
27-Jun-2021 | clsneikex 37424 | If closure and neighborhoods functions are related, the closure function exists. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) & ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜)))))) & ⊢ 𝐷 = (𝑃‘𝐵) & ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) & ⊢ 𝐻 = (𝐹 ∘ 𝐷) & ⊢ (𝜑 → 𝐾𝐻𝑁) ⇒ ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) | ||
27-Jun-2021 | ntrclsss 37381 | If interior and closure functions are related then a subset relation of a pair of function values is equivalent to subset relation of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) & ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆)))) | ||
27-Jun-2021 | ntrclsfveq 37380 | If interior and closure functions are related then equality of a pair of function values is equivalent to equality of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) & ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇)))) | ||
27-Jun-2021 | ntrclsfveq2 37379 | If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐾‘𝑆) = (𝐵 ∖ 𝐶))) | ||
27-Jun-2021 | ntrclsfveq1 37378 | If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.) |
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) & ⊢ 𝐷 = (𝑂‘𝐵) & ⊢ (𝜑 → 𝐼𝐷𝐾) & ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶))) | ||
26-Jun-2021 | smfco 39687 | The composition of a Borel sigma-measurable function with a sigma-measurable function, is sigma-measurable. Proposition 121E (g) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐻 ∈ (SMblFn‘𝐵)) ⇒ ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smf2id 39686 | Twice the identity function is Borel sigma-measurable (just an example, to test previous general theorems). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (2 · 𝑥)) ∈ (SMblFn‘𝐵)) | ||
26-Jun-2021 | smfpimbor1 39685 | Given a sigma-measurable function, the preimage of a Borel set belongs to the subspace sigma-algebra induced by the domain of the function. Proposition 121E (f) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ 𝑃 = (◡𝐹 “ 𝐸) ⇒ ⊢ (𝜑 → 𝑃 ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | smfpimbor1lem2 39684 | Given D sigma-measurable function, the preimage of D Borel set belongs to the subspace sigma-algebra induced by the domain of the function. Proposition 121E (f) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ 𝑃 = (◡𝐹 “ 𝐸) & ⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)} ⇒ ⊢ (𝜑 → 𝑃 ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | smfpimbor1lem1 39683 | Every open set belongs to 𝑇. This is the second step in the proof of Proposition 121E (f) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐺 ∈ 𝐽) & ⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)} ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝑇) | ||
26-Jun-2021 | smfdiv 39682 | The fraction of two sigma-measurable functions is measurable. Proposition 121E (e) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) & ⊢ 𝐸 = {𝑥 ∈ 𝐶 ∣ 𝐷 ≠ 0} ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐸) ↦ (𝐵 / 𝐷)) ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smfmulc1 39681 | A sigma-measurable function multiplied by a constant, is sigma-measurable. Proposition 121E (c) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smfmul 39680 | The multiplication of two sigma-measurable functions is measurable. Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 · 𝐷)) ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smfmullem4 39679 | The multiplication of two sigma-measurable functions is measurable. Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ 𝐾 = {𝑞 ∈ (ℚ ↑𝑚 (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑅} & ⊢ 𝐸 = (𝑞 ∈ 𝐾 ↦ {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝐷 ∈ ((𝑞‘2)(,)(𝑞‘3)))}) ⇒ ⊢ (𝜑 → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 · 𝐷) < 𝑅} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) | ||
26-Jun-2021 | smfmullem3 39678 | The multiplication of two sigma-measurable functions is measurable: this is the step (i) of the proof of Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ 𝐾 = {𝑞 ∈ (ℚ ↑𝑚 (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑅} & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝑉 ∈ ℝ) & ⊢ (𝜑 → (𝑈 · 𝑉) < 𝑅) & ⊢ 𝑋 = ((𝑅 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉)))) & ⊢ 𝑌 = if(1 ≤ 𝑋, 1, 𝑋) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐾 (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3)))) | ||
26-Jun-2021 | smfmullem2 39677 | The multiplication of two sigma-measurable functions is measurable: this is the step (i) of the proof of Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝐾 = {𝑞 ∈ (ℚ ↑𝑚 (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝐴} & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝑉 ∈ ℝ) & ⊢ (𝜑 → (𝑈 · 𝑉) < 𝐴) & ⊢ (𝜑 → 𝑃 ∈ ℚ) & ⊢ (𝜑 → 𝑅 ∈ ℚ) & ⊢ (𝜑 → 𝑆 ∈ ℚ) & ⊢ (𝜑 → 𝑍 ∈ ℚ) & ⊢ (𝜑 → 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈)) & ⊢ (𝜑 → 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌))) & ⊢ (𝜑 → 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉)) & ⊢ (𝜑 → 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌))) & ⊢ 𝑋 = ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉)))) & ⊢ 𝑌 = if(1 ≤ 𝑋, 1, 𝑋) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐾 (𝑈 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝑉 ∈ ((𝑞‘2)(,)(𝑞‘3)))) | ||
26-Jun-2021 | smfmullem1 39676 | The multiplication of two sigma-measurable functions is measurable: this is the step (i) of the proof of Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑈 ∈ ℝ) & ⊢ (𝜑 → 𝑉 ∈ ℝ) & ⊢ (𝜑 → (𝑈 · 𝑉) < 𝐴) & ⊢ 𝑋 = ((𝐴 − (𝑈 · 𝑉)) / (1 + ((abs‘𝑈) + (abs‘𝑉)))) & ⊢ 𝑌 = if(1 ≤ 𝑋, 1, 𝑋) & ⊢ (𝜑 → 𝑃 ∈ ((𝑈 − 𝑌)(,)𝑈)) & ⊢ (𝜑 → 𝑅 ∈ (𝑈(,)(𝑈 + 𝑌))) & ⊢ (𝜑 → 𝑆 ∈ ((𝑉 − 𝑌)(,)𝑉)) & ⊢ (𝜑 → 𝑍 ∈ (𝑉(,)(𝑉 + 𝑌))) & ⊢ (𝜑 → 𝐻 ∈ (𝑃(,)𝑅)) & ⊢ (𝜑 → 𝐼 ∈ (𝑆(,)𝑍)) ⇒ ⊢ (𝜑 → (𝐻 · 𝐼) < 𝐴) | ||
26-Jun-2021 | smfres 39675 | The restriction of sigma-measurable function is sigma-measurable. Proposition 121E (h) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smfrec 39674 | The reciprocal of a sigma-measurable functions is sigma-measurable. First part of Proposition 121E (e) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 0} ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (1 / 𝐵)) ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smfresal 39673 | Given a sigma-measurable function, the subsets of ℝ whose preimage is in the sigma-algebra induced by the function's domain, form a sigma-algebra. First part of the proof of Proposition 121E (f) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)} ⇒ ⊢ (𝜑 → 𝑇 ∈ SAlg) | ||
26-Jun-2021 | smfpimioo 39672 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (◡𝐹 “ (𝐴(,)𝐵)) ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | smfpimioompt 39671 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐿 ∈ ℝ*) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} ∈ (𝑆 ↾t 𝐴)) | ||
26-Jun-2021 | smfpimgtxrmpt 39670 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐿 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾t 𝐴)) | ||
26-Jun-2021 | mbfpsssmf 39669 | Real valued, measurable functions are a proper subset of sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑆 = dom vol ⇒ ⊢ (MblFn ∩ (ℝ ↑pm ℝ)) ⊊ (SMblFn‘𝑆) | ||
26-Jun-2021 | smfpreimage 39668 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of a closed interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | smfpimgtmpt 39667 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐿 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾t 𝐴)) | ||
26-Jun-2021 | smfpimgtxr 39666 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | nsssmfmbf 39665 | The sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals) are not a subset of the measurable functions. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑆 = dom vol ⇒ ⊢ ¬ (SMblFn‘𝑆) ⊆ MblFn | ||
26-Jun-2021 | nsssmfmbflem 39664 | The sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals) are not a subset of the measurable functions. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑆 = dom vol & ⊢ (𝜑 → 𝑋 ⊆ ℝ) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 0) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓 ∈ (SMblFn‘𝑆) ∧ ¬ 𝑓 ∈ MblFn)) | ||
26-Jun-2021 | smflim 39663 | The limit of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . Notice that every function in the sequence can have a different (partial) domain, and the domain of convergence can be decidedly irregular (Remark 121G of [Fremlin1] p. 39 ). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smflimlem6 39662 | Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | smflimlem5 39661 | Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) & ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))) & ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) & ⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | smflimlem4 39660 | Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves one-side of the double inclusion for the proof that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) & ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))) & ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) & ⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟) ⇒ ⊢ (𝜑 → (𝐷 ∩ 𝐼) ⊆ {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴}) | ||
26-Jun-2021 | smflimlem3 39659 | The limit of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) & ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))) & ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) & ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑃) → (𝐶‘𝑦) ∈ 𝑦) & ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∩ 𝐼)) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝑌 ∈ ℝ+) & ⊢ (𝜑 → (1 / 𝐾) < 𝑌) ⇒ ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + 𝑌))) | ||
26-Jun-2021 | smflimlem2 39658 | Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves one-side of the double inclusion for the proof that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) & ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))) & ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) & ⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ⊆ (𝐷 ∩ 𝐼)) | ||
26-Jun-2021 | smflimlem1 39657 | Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves that (𝐷 ∩ 𝐼) is in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) & ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))) & ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚𝐻𝑘) & ⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟) ⇒ ⊢ (𝜑 → (𝐷 ∩ 𝐼) ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | issmfge 39656 | The predicate "𝐹 is b measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-closed intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iv) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)))) | ||
26-Jun-2021 | issmfgelem 39655 | The predicate "𝐹 is a measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-closed intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iv) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 ≤ (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smfpreimagtf 39654 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | decsmf 39653 | A real valued, non-increasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) | ||
26-Jun-2021 | decsmflem 39652 | A non-increasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} & ⊢ 𝐶 = sup(𝑌, ℝ*, < ) & ⊢ 𝐷 = (-∞(,)𝐶) & ⊢ 𝐸 = (-∞(,]𝐶) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) | ||
26-Jun-2021 | smfadd 39651 | The sum of two sigma-measurable functions is measurable. Proposition 121E (b) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)) ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smfaddlem2 39650 | The sum of two sigma-measurable functions is measurable. Proposition 121E (b) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ 𝐾 = (𝑝 ∈ ℚ ↦ {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅}) ⇒ ⊢ (𝜑 → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 + 𝐷) < 𝑅} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) | ||
26-Jun-2021 | smfaddlem1 39649 | Given the sum of two functions, the preimage of an unbounded below, open interval, expressed as the countable union of intersections of preimages of both functions. Proposition 121E (b) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ 𝐾 = (𝑝 ∈ ℚ ↦ {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑅}) ⇒ ⊢ (𝜑 → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 + 𝐷) < 𝑅} = ∪ 𝑝 ∈ ℚ ∪ 𝑞 ∈ (𝐾‘𝑝){𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 < 𝑝 ∧ 𝐷 < 𝑞)}) | ||
26-Jun-2021 | smfpreimagt 39648 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | issmfgtd 39647 | A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smfmbfcex 39646 | A constant function, with non-lebesgue-measurable domain is a sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals) but it is not a measurable functions ( w.r.t. to df-mbf 23194). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑆 = dom vol & ⊢ (𝜑 → 𝑋 ⊆ ℝ) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 0) ⇒ ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ∧ ¬ 𝐹 ∈ MblFn)) | ||
26-Jun-2021 | smfpimltxrmpt 39645 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾t 𝐴)) | ||
26-Jun-2021 | issmfled 39644 | A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | issmfgt 39643 | The predicate "𝐹 is b measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-open intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)))) | ||
26-Jun-2021 | issmfgtlem 39642 | The predicate "𝐹 is a measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-open intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smfpreimale 39641 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an closed interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | bormflebmf 39640 | A Borel measurable function is Lebesgue measurable. Proposition 121D (a) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ 𝐵 = (SalGen‘(TopOpen‘(ℝ^‘𝑋))) & ⊢ 𝐿 = dom (voln‘𝑋) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐿)) | ||
26-Jun-2021 | smfid 39639 | The identity function is Borel sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (SMblFn‘𝐵)) | ||
26-Jun-2021 | cnfrrnsmf 39638 | A function, continuous from the standard topology on the space of n-dimensional reals, and the standard topology on the reals, is Borel measurable. Proposition 121D (b) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ 𝐽 = (TopOpen‘(ℝ^‘𝑋)) & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾)) & ⊢ 𝐵 = (SalGen‘𝐽) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) | ||
26-Jun-2021 | sssmfmpt 39637 | The restriction of a sigma-measurable function, is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smfconst 39636 | A constant function is measurable. Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | issmfdmpt 39635 | A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑎} ∈ (𝑆 ↾t 𝐴)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smfpimltxr 39634 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | smfpimltmpt 39633 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝑅 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾t 𝐴)) | ||
26-Jun-2021 | issmfle 39632 | The predicate "𝐹 is b measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all right closed intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (ii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)))) | ||
26-Jun-2021 | issmflelem 39631 | The predicate "𝐹 is a measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all right closed intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (ii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smfsssmf 39630 | If a function is measurable w.r.t. to a sigma-algebra, then it is measurable w.r.t. to a larger sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑅 ∈ SAlg) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝑅 ⊆ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑅)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | incsmf 39629 | A real valued, non-decreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) | ||
26-Jun-2021 | incsmflem 39628 | A non decreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} & ⊢ 𝐶 = sup(𝑌, ℝ*, < ) & ⊢ 𝐷 = (-∞(,)𝐶) & ⊢ 𝐸 = (-∞(,]𝐶) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) | ||
26-Jun-2021 | cnfsmf 39627 | A continuous function is measurable. Proposition 121D (b) of [Fremlin1] p. 36 is a special case of this theorem, where the topology on the domain is induced by the standard topology on n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾)) & ⊢ 𝑆 = (SalGen‘𝐽) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | mbfresmf 39626 | A Real valued, measurable function is a sigma-measurable function (w.r.t. the Lebesgue measure on the Reals). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) & ⊢ 𝑆 = dom vol ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | sssmf 39625 | The restriction of a sigma-measurable function, is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | issmfdf 39624 | A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | smfpreimaltf 39623 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | issmfltle 39622 | The definition of a measurable function w.r.t. a sigma-algebra, can be stated using less than or equal instead of less than. Proposition 121B (ii) of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | issmfd 39621 | A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
26-Jun-2021 | issmff 39620 | The predicate "𝐹 is a measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) | ||
26-Jun-2021 | smfdmss 39619 | The domain of a function measurable w.r.t. to a sigma-algebra, is a subset of the set underlying the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) | ||
26-Jun-2021 | smff 39618 | A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) | ||
26-Jun-2021 | smfpreimalt 39617 | Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)) | ||
26-Jun-2021 | salpreimagtlt 39616 | If all the preimages of lef-open, unbounded above intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (iii) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐴 = ∪ 𝑆 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) | ||
26-Jun-2021 | salpreimalelt 39615 | If all the preimages of right-close, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (ii) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐴 = ∪ 𝑆 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) | ||
26-Jun-2021 | issmf 39614 | The predicate "𝐹 is a measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) | ||
26-Jun-2021 | issmflem 39613 | The predicate "𝐹 is a measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) | ||
26-Jun-2021 | salpreimaltle 39612 | If all the preimages of right-open, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-closed, unbounded below intervals, belong to the sigma-algebra. (i) implies (ii) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑎} ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆) | ||
26-Jun-2021 | salpreimagtge 39611 | If all the preimages of left-open, unbounded above intervals, belong to a sigma-algebra, then all the preimages of left-closed, unbounded above intervals, belong to the sigma-algebra. (iii) implies (iv) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆) | ||
26-Jun-2021 | pimrecltneg 39610 | The preimage of an unbounded below, open interval, with negative upper bound, for the reciprocal function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ 0) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐶 < 0) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (1 / 𝐵) < 𝐶} = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ ((1 / 𝐶)(,)0)}) | ||
26-Jun-2021 | pimgtmnf 39609 | Given a real valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴) | ||
26-Jun-2021 | preimaleiinlt 39608 | A preimage of a left-open, right-closed, unbounded below interval, expressed as an indexed intersection of preimages of open, unbound below intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} = ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐴 ∣ 𝐵 < (𝐶 + (1 / 𝑛))}) | ||
26-Jun-2021 | preimageiingt 39607 | A preimage of a left-closed, unbounded above interval, expressed as an indexed intersection of preimages of open, unbounded above intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} = ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵}) | ||
26-Jun-2021 | preimaioomnf 39606 | Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) | ||
26-Jun-2021 | pimincfltioo 39605 | Given a non decreasing function, the preimage of an unbounded below, open interval, when the supremum of the preimage does not belong to the preimage. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} & ⊢ 𝑆 = sup(𝑌, ℝ*, < ) & ⊢ (𝜑 → ¬ 𝑆 ∈ 𝑌) & ⊢ 𝐼 = (-∞(,)𝑆) ⇒ ⊢ (𝜑 → 𝑌 = (𝐼 ∩ 𝐴)) | ||
26-Jun-2021 | pimdecfgtioo 39604 | Given a non decreasing function, the preimage of an unbounded below, open interval, when the supremum of the preimage does not belong to the preimage. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} & ⊢ 𝑆 = sup(𝑌, ℝ*, < ) & ⊢ (𝜑 → ¬ 𝑆 ∈ 𝑌) & ⊢ 𝐼 = (-∞(,)𝑆) ⇒ ⊢ (𝜑 → 𝑌 = (𝐼 ∩ 𝐴)) | ||
26-Jun-2021 | pimincfltioc 39603 | Given a non decreasing function, the preimage of an unbounded below, open interval, when the supremum of the preimage belongs to the preimage. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} & ⊢ 𝑆 = sup(𝑌, ℝ*, < ) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) & ⊢ 𝐼 = (-∞(,]𝑆) ⇒ ⊢ (𝜑 → 𝑌 = (𝐼 ∩ 𝐴)) | ||
26-Jun-2021 | pimdecfgtioc 39602 | Given a non-increasing function, the preimage of an unbounded above, open interval, when the supremum of the preimage belongs to the preimage. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} & ⊢ 𝑆 = sup(𝑌, ℝ*, < ) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) & ⊢ 𝐼 = (-∞(,]𝑆) ⇒ ⊢ (𝜑 → 𝑌 = (𝐼 ∩ 𝐴)) | ||
26-Jun-2021 | pimgtmnf2 39601 | Given a real valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} = 𝐴) | ||
26-Jun-2021 | pimltpnf2 39600 | Given a real valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴) | ||
26-Jun-2021 | preimaicomnf 39599 | Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) | ||
26-Jun-2021 | pimiooltgt 39598 | The preimage of an open interval is the intersection of the preimage of an unbounded below open interval and an unbounded above open interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐿 ∈ ℝ*) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∩ {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵})) | ||
26-Jun-2021 | salpreimalegt 39597 | If all the preimages of right-closed, unbounded below intervals, belong to a sigma-algebra, then all the preimages of left-open, unbounded above intervals, belong to the sigma-algebra. (ii) implies (iii) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐴 = ∪ 𝑆 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ 𝑆) | ||
26-Jun-2021 | pimrecltpos 39596 | The preimage of an unbounded below, open interval, with positive upper bound, for the reciprocal function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ 0) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (1 / 𝐵) < 𝐶} = ({𝑥 ∈ 𝐴 ∣ (1 / 𝐶) < 𝐵} ∪ {𝑥 ∈ 𝐴 ∣ 𝐵 < 0})) | ||
26-Jun-2021 | salpreimagelt 39595 | If all the preimages of left-close, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (iv) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ 𝐴 = ∪ 𝑆 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) | ||
26-Jun-2021 | pimgtpnf2 39594 | Given a real valued function, the preimage of an open interval, unbounded above, with lower bound +∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅) | ||
26-Jun-2021 | pimltpnf 39593 | Given a real valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} = 𝐴) | ||
26-Jun-2021 | pimconstlt1 39592 | Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound larger than the constant, is the whole domain. First part of Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = 𝐴) | ||
26-Jun-2021 | pimconstlt0 39591 | Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound smaller or equal to the constant, is the empty set. Second part of Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ≤ 𝐵) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅) | ||
26-Jun-2021 | preimalegt 39590 | The preimage of a left-open, unbounded above interval, is the complement of a right-close, unbounded below interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) | ||
26-Jun-2021 | preimagelt 39589 | The preimage of a right-open, unbounded below interval, is the complement of a left-close, unbounded above interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) | ||
26-Jun-2021 | pimltmnf2 39588 | Given a real valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅) | ||
26-Jun-2021 | df-smblfn 39587 | Define a measurable function w.r.t. a given sigma-algebra. See Definition 121C of [Fremlin1] p. 36 and Definition 135E (b) of [Fremlin1] p. 80 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)}) | ||
26-Jun-2021 | vitali2 39585 | There are non-measurable sets (the Axiom of Choice is used, in the invoked weth 9200). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ dom vol ⊊ 𝒫 ℝ | ||
26-Jun-2021 | subsaluni 39254 | A set belongs to the subspace sigma-algebra it induces. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝑆 ↾t 𝐴)) | ||
26-Jun-2021 | subsalsal 39253 | A subspace sigma-algebra is a sigma algebra. This is Lemma 121A of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑇 = (𝑆 ↾t 𝐷) ⇒ ⊢ (𝜑 → 𝑇 ∈ SAlg) | ||
26-Jun-2021 | subsaliuncl 39252 | A subspace sigma-algebra is closed under countable union. This is Lemma 121A (iii) of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑇 = (𝑆 ↾t 𝐷) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑇) ⇒ ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇) | ||
26-Jun-2021 | subsaliuncllem 39251 | A subspace sigma-algebra is closed under countable union. This is Lemma 121A (iii) of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) & ⊢ 𝐸 = (𝐻 ∘ 𝐺) & ⊢ (𝜑 → 𝐻 Fn ran 𝐺) & ⊢ (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑦) ⇒ ⊢ (𝜑 → ∃𝑒 ∈ (𝑆 ↑𝑚 ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) | ||
26-Jun-2021 | iocborel 39250 | A left-open, right-closed interval is a Borel set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) ⇒ ⊢ (𝜑 → (𝐴(,]𝐶) ∈ 𝐵) | ||
26-Jun-2021 | bor1sal 39249 | The Borel sigma-algebra on the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) ⇒ ⊢ 𝐵 ∈ SAlg | ||
26-Jun-2021 | unisalgen2 39248 | The union of a set belongs is equal to the union of the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ 𝑆 = (SalGen‘𝐴) ⇒ ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐴) | ||
26-Jun-2021 | salunid 39247 | A set is an element of any sigma-algebra on it . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) ⇒ ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆) | ||
26-Jun-2021 | salincld 39246 | The intersection of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐸 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐸 ∩ 𝐹) ∈ 𝑆) | ||
26-Jun-2021 | iooborel 39245 | An open interval is a Borel set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) ⇒ ⊢ (𝐴(,)𝐶) ∈ 𝐵 | ||
26-Jun-2021 | 0sald 39244 | The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) ⇒ ⊢ (𝜑 → ∅ ∈ 𝑆) | ||
26-Jun-2021 | salgencld 39243 | SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ 𝑆 = (SalGen‘𝑋) ⇒ ⊢ (𝜑 → 𝑆 ∈ SAlg) | ||
26-Jun-2021 | saluncld 39242 | The union of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐸 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝑆) | ||
26-Jun-2021 | saldifcld 39241 | The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐸 ∈ 𝑆) ⇒ ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) | ||
26-Jun-2021 | mbfdmssre 38893 | The domain of a measurable function is a subset of the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ) | ||
26-Jun-2021 | fnlimabslt 38746 | A sequence of function values, approximates the corresponding limit function value, all but finitely many times. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)(((𝐹‘𝑚)‘𝑋) ∈ ℝ ∧ (abs‘(((𝐹‘𝑚)‘𝑋) − (𝐺‘𝑋))) < 𝑌)) | ||
26-Jun-2021 | fnlimf 38745 | The limit function of real functions, is a real valued function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺:𝐷⟶ℝ) | ||
26-Jun-2021 | fnlimfvre2 38744 | The limit function of real functions, applied to elements in its domain, evaluates to Real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝐺‘𝑋) ∈ ℝ) | ||
26-Jun-2021 | climleltrp 38743 | The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))) | ||
26-Jun-2021 | allbutfifvre 38742 | Given a sequence of real valued functions, and 𝑋 that belongs to all but finitely many domains, then its function value is ultimately a real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑚𝜑 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) & ⊢ 𝐷 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ) | ||
26-Jun-2021 | fnlimfvre 38741 | The limit function of real functions, applied to elements in its domain, evaluates to Real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) | ||
26-Jun-2021 | clim2d 38740 | The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋)) | ||
26-Jun-2021 | climd 38739 | Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋)) | ||
26-Jun-2021 | climfveqmpt 38738 | Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑅) & ⊢ (𝜑 → 𝑍 ⊆ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ (𝜑 → 𝑍 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐷 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → ( ⇝ ‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = ( ⇝ ‘(𝑘 ∈ 𝐶 ↦ 𝐷))) | ||
26-Jun-2021 | climfveq 38736 | Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) | ||
26-Jun-2021 | climeldmeqmpt 38735 | Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑅) & ⊢ (𝜑 → 𝑍 ⊆ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ (𝜑 → 𝑍 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐷 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ dom ⇝ ↔ (𝑘 ∈ 𝐶 ↦ 𝐷) ∈ dom ⇝ )) | ||
26-Jun-2021 | fnlimcnv 38734 | The sequence of function values converges to the value of the limit function 𝐺 at any point of its domain 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐹 & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ (𝐺‘𝑋)) | ||
26-Jun-2021 | climf2 38733 | Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. Similar to clim 14073, but without the disjoint var constraint 𝜑𝑘 and 𝐹𝑘. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))) | ||
26-Jun-2021 | climeldmeq 38732 | Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) | ||
26-Jun-2021 | climreclf 38731 | The limit of a convergent real sequence is real. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
26-Jun-2021 | fnlimfv 38730 | The value of the limit function 𝐺 at any point of its domain 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐷 & ⊢ Ⅎ𝑥𝐹 & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝐺‘𝑋) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))) | ||
26-Jun-2021 | ressiooinf 38631 | If the infimum does not belong to a set of reals, the set is a subset of the unbounded above, left-open interval, with lower bound equal to the infimum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑆 = inf(𝐴, ℝ*, < ) & ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) & ⊢ 𝐼 = (𝑆(,)+∞) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐼) | ||
26-Jun-2021 | iooiinioc 38630 | A left-open, right-closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,]𝐵)) | ||
26-Jun-2021 | ressioosup 38629 | If the supremum does not belong to a set of reals, the set is a subset of the unbounded below, right-open interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑆 = sup(𝐴, ℝ*, < ) & ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) & ⊢ 𝐼 = (-∞(,)𝑆) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐼) | ||
26-Jun-2021 | ressiocsup 38628 | If the supremum belongs to a set of reals, the set is a subset of the unbounded below, right-closed interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑆 = sup(𝐴, ℝ*, < ) & ⊢ (𝜑 → 𝑆 ∈ 𝐴) & ⊢ 𝐼 = (-∞(,]𝑆) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐼) | ||
26-Jun-2021 | sqrlearg 38627 | The square compared with its argument. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴↑2) ≤ 𝐴 ↔ 𝐴 ∈ (0[,]1))) | ||
26-Jun-2021 | icomnfinre 38626 | A left-closed, right-open, interval of extended reals, intersected with the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴)) | ||
26-Jun-2021 | ioofun 38625 | (,) is a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Fun (,) | ||
26-Jun-2021 | ioogtlbd 38624 | An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
26-Jun-2021 | elioored 38623 | A member of an open interval of reals is a real. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ (𝐵(,)𝐶)) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
26-Jun-2021 | iccleubd 38622 | An element of a closed interval is less than or equal to its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) ⇒ ⊢ (𝜑 → 𝐶 ≤ 𝐵) | ||
26-Jun-2021 | tgqioo2 38621 | Every open set of reals is the (countable) union of open interval with rational bounds. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) ⇒ ⊢ (𝜑 → ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐴 = ∪ 𝑞)) | ||
26-Jun-2021 | qelioo 38620 | The rational numbers are dense in ℝ*: any two extended real numbers have a rational between them. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℚ 𝑥 ∈ (𝐴(,)𝐵)) | ||
26-Jun-2021 | icoltubd 38619 | An element of a left closed right open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) ⇒ ⊢ (𝜑 → 𝐶 < 𝐵) | ||
26-Jun-2021 | iooltubd 38618 | An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) ⇒ ⊢ (𝜑 → 𝐶 < 𝐵) | ||
26-Jun-2021 | iccgelbd 38617 | An element of a closed interval is more than or equal to its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐶) | ||
26-Jun-2021 | ltdiv23neg 38558 | Swap denominator with other side of 'less than', when both are negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐵 < 0) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐶 < 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵)) | ||
26-Jun-2021 | allbutfi 38557 | For all but finitely many. Some authors say "cofinitely many". Some authors say "ultimately". Compare with eliuniin 38307 and eliuniin2 38335 (here, the precondition can be dropped; see eliuniincex 38323). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐴 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 ⇒ ⊢ (𝑋 ∈ 𝐴 ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵) | ||
26-Jun-2021 | ltmulneg 38556 | Multiplying by a negative number, swaps the order. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐶 < 0) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐵 · 𝐶) < (𝐴 · 𝐶))) | ||
26-Jun-2021 | reclt0 38555 | The reciprocal of a negative number is negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 < 0 ↔ (1 / 𝐴) < 0)) | ||
26-Jun-2021 | xrralrecnnge 38554 | Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (𝐴 − (1 / 𝑛)) ≤ 𝐵)) | ||
26-Jun-2021 | infxrcld 38553 | The infimum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ*) ⇒ ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ∈ ℝ*) | ||
26-Jun-2021 | negelrpd 38552 | The negation of a negative number is in the positive real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 0) ⇒ ⊢ (𝜑 → -𝐴 ∈ ℝ+) | ||
26-Jun-2021 | xrleidd 38551 | 'Less than or equal to' is reflexive for extended reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐴) | ||
26-Jun-2021 | mnfled 38550 | Minus infinity is less than or equal to any extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → -∞ ≤ 𝐴) | ||
26-Jun-2021 | lt0neg1dd 38549 | If a number is negative, its negative is positive. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 0) ⇒ ⊢ (𝜑 → 0 < -𝐴) | ||
26-Jun-2021 | reclt0d 38548 | The reciprocal of a negative number is negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 0) ⇒ ⊢ (𝜑 → (1 / 𝐴) < 0) | ||
26-Jun-2021 | qred 38547 | A rational number is a real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℚ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
26-Jun-2021 | nnrecrp 38546 | The reciprocal of a positive natural number is a positive real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ+) | ||
26-Jun-2021 | frexr 38545 | A function taking real values, is a function taking extended real values. Common case. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | ||
26-Jun-2021 | axccd2 38425 | An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) ⇒ ⊢ (𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) | ||
26-Jun-2021 | axccd 38424 | An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ≈ ω) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) ⇒ ⊢ (𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) | ||
26-Jun-2021 | elrnmpt2id 38422 | Membership in the range of an operation class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) ∈ ran 𝐹) | ||
26-Jun-2021 | frnd 38421 | The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵) | ||
26-Jun-2021 | freld 38420 | A mapping is a relation. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → Rel 𝐹) | ||
26-Jun-2021 | fcod 38419 | Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | ||
26-Jun-2021 | fvcod 38418 | Value of a function composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → Fun 𝐺) & ⊢ (𝜑 → 𝐴 ∈ dom 𝐺) & ⊢ 𝐻 = (𝐹 ∘ 𝐺) ⇒ ⊢ (𝜑 → (𝐻‘𝐴) = (𝐹‘(𝐺‘𝐴))) | ||
26-Jun-2021 | dmexd 38417 | The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → dom 𝐴 ∈ V) | ||
26-Jun-2021 | fco3 38416 | Functionality of a composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → Fun 𝐺) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹) | ||
26-Jun-2021 | fdmd 38415 | The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → dom 𝐹 = 𝐴) | ||
26-Jun-2021 | dmrelrnrel 38414 | A relation preserving function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → (𝐹‘𝐵)𝑆(𝐹‘𝐶)) | ||
26-Jun-2021 | elpmi2 38413 | The domain of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → dom 𝐹 ⊆ 𝐵) | ||
26-Jun-2021 | dmmptdf 38412 | The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → dom 𝐴 = 𝐵) | ||
26-Jun-2021 | axccdom 38411 | Relax the constraint on ax-cc to dominance instead of equinumerosity. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑋 ≼ ω) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ≠ ∅) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧 ∈ 𝑋 (𝑓‘𝑧) ∈ 𝑧)) | ||
26-Jun-2021 | imaexi 38410 | The image of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐴 ∈ 𝑉 ⇒ ⊢ (𝐴 “ 𝐵) ∈ V | ||
26-Jun-2021 | elpmrn 38409 | The range of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → ran 𝐹 ⊆ 𝐴) | ||
26-Jun-2021 | unirestss 38339 | The union of an elementwise intersection is a subset of the underlying set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) ⊆ ∪ 𝐴) | ||
26-Jun-2021 | restuni5 38338 | The underlying set of a subspace induced by the ↾t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) | ||
26-Jun-2021 | restuni6 38337 | The underlying set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵)) | ||
26-Jun-2021 | restuni4 38336 | The underlying set of a subspace induced by the ↾t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴) ⇒ ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵) | ||
26-Jun-2021 | eliuniin2 38335 | Indexed union of indexed intersections. See eliincex 38324 for a counterexample showing that the precondition 𝐶 ≠ ∅ cannot be simply dropped. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐶 & ⊢ 𝐴 = ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷 ⇒ ⊢ (𝐶 ≠ ∅ → (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)) | ||
26-Jun-2021 | rabssf 38334 | Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) | ||
26-Jun-2021 | restuni3 38333 | The underlying set of a subspace induced by the subspace operator ↾t. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵)) | ||
26-Jun-2021 | rabeqi 38332 | Equality theorem for restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} | ||
26-Jun-2021 | ssrab2f 38331 | Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | ||
26-Jun-2021 | eliin2 38330 | Membership in indexed intersection. See eliincex 38324 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4461 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 38316). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) | ||
26-Jun-2021 | ssrabf 38329 | Subclass of a restricted class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑)) | ||
26-Jun-2021 | supxrubd 38328 | A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ 𝑆 = sup(𝐴, ℝ*, < ) ⇒ ⊢ (𝜑 → 𝐵 ≤ 𝑆) | ||
26-Jun-2021 | fexd 38327 | If the domain of a mapping is a set, the function is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐹 ∈ V) | ||
26-Jun-2021 | abssf 38326 | Class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) | ||
26-Jun-2021 | eliinid 38325 | Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶) | ||
26-Jun-2021 | eliincex 38324 | Counterexample to show that the additional conditions in eliin 4461 and eliin2 38330 are actually needed. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐴 = V & ⊢ 𝐵 = ∅ ⇒ ⊢ ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) |
(29-Jul-2020) Mario Carneiro presented MM0 at the CICM conference. See this Google Group post which includes a YouTube link.
(20-Jul-2020) Rohan Ridenour found 5 shorter D-proofs in our Shortest known proofs... file. In particular, he reduced *4.39 from 901 to 609 steps. A note on the Metamath Solitaire page mentions a tool that he worked with.
(19-Jul-2020) David A. Wheeler posted a video (https://youtu.be/3R27Qx69jHc) on how to (re)prove Schwabhäuser 4.6 for the Metamath Proof Explorer. See also his older videos.
(19-Jul-2020) In version 0.184 of the metamath program, "verify markup" now checks that mathboxes are independent i.e. do not cross-reference each other. To turn off this check, use "/mathbox_skip"
(30-Jun-2020) In version 0.183 of the metamath program, (1) "verify markup" now has checking for (i) underscores in labels, (ii) that *ALT and *OLD theorems have both discouragement tags, and (iii) that lines don't have trailing spaces. (2) "save proof.../rewrap" no longer left-aligns $p/$a comments that contain the string "<HTML>"; see this note.
(5-Apr-2020) Glauco Siliprandi added a new proof to the 100 theorem list, e is Transcendental etransc, bringing the Metamath total to 74.
(12-Feb-2020) A bug in the 'minimize' command of metamath.exe versions 0.179 (29-Nov-2019) and 0.180 (10-Dec-2019) may incorrectly bring in the use of new axioms. Version 0.181 fixes it.
(20-Jan-2020) David A. Wheeler created a video called Walkthrough of the tutorial in mmj2. See the Google Group announcement for more details. (All of his videos are listed on the Other Metamath-Related Topics page.)
(18-Jan-2020) The FOMM 2020 talks are on youtube now. Mario Carneiro's talk is Metamath Zero, or: How to Verify a Verifier. Since they are washed out in the video, the PDF slides are available separately.
(14-Dec-2019) Glauco Siliprandi added a new proof to the 100 theorem list, Fourier series convergence fourier, bringing the Metamath total to 73.
(25-Nov-2019) Alexander van der Vekens added a new proof to the 100 theorem list, The Cayley-Hamilton Theorem cayleyhamilton, bringing the Metamath total to 72.
(25-Oct-2019) Mario Carneiro's paper "Metamath Zero: The Cartesian Theorem Prover" (submitted to CPP 2020) is now available on arXiv: https://arxiv.org/abs/1910.10703. There is a related discussion on Hacker News.
(30-Sep-2019) Mario Carneiro's talk about MM0 at ITP 2019 is available on YouTube: x86 verification from scratch (24 minutes). Google Group discussion: Metamath Zero.
(29-Sep-2019) David Wheeler created a fascinating Gource video that animates the construction of set.mm, available on YouTube: Metamath set.mm contributions viewed with Gource through 2019-09-26 (4 minutes). Google Group discussion: Gource video of set.mm contributions.
(24-Sep-2019) nLab added a page for Metamath. It mentions Stefan O'Rear's Busy Beaver work using the set.mm axiomatization (and fails to mention Mario's definitional soundness checker)
(1-Sep-2019) Xuanji Li published a Visual Studio Code extension to support metamath syntax highlighting.
(10-Aug-2019) (revised 21-Sep-2019) Version 0.178 of the metamath program has the following changes: (1) "minimize_with" will now prevent dependence on new $a statements unless the new qualifier "/allow_new_axioms" is specified. For routine usage, it is suggested that you use "minimize_with * /allow_new_axioms * /no_new_axioms_from ax-*" instead of just "minimize_with *". See "help minimize_with" and this Google Group post. Also note that the qualifier "/allow_growth" has been renamed to "/may_grow". (2) "/no_versioning" was added to "write theorem_list".
(8-Jul-2019) Jon Pennant announced the creation of a Metamath search engine. Try it and feel free to comment on it at https://groups.google.com/d/msg/metamath/cTeU5AzUksI/5GesBfDaCwAJ.
(16-May-2019) Set.mm now has a major new section on elementary geometry. This begins with definitions that implement Tarski's axioms of geometry (including concepts such as congruence and betweenness). This uses set.mm's extensible structures, making them easier to use for many circumstances. The section then connects Tarski geometry with geometry in Euclidean places. Most of the work in this section is due to Thierry Arnoux, with earlier work by Mario Carneiro and Scott Fenton. [Reported by DAW.]
(9-May-2019) We are sad to report that long-time contributor Alan Sare passed away on Mar. 23. There is some more information at the top of his mathbox (click on "Mathbox for Alan Sare") and his obituary. We extend our condolences to his family.
(10-Mar-2019) Jon Pennant and Mario Carneiro added a new proof to the 100 theorem list, Heron's formula heron, bringing the Metamath total to 71.
(22-Feb-2019) Alexander van der Vekens added a new proof to the 100 theorem list, Cramer's rule cramer, bringing the Metamath total to 70.
(6-Feb-2019) David A. Wheeler has made significant improvements and updates to the Metamath book. Any comments, errors found, or suggestions are welcome and should be turned into an issue or pull request at https://github.com/metamath/metamath-book (or sent to me if you prefer).
(26-Dec-2018) I added Appendix 8 to the MPE Home Page that cross-references new and old axiom numbers.
(20-Dec-2018) The axioms have been renumbered according to this Google Groups post.
(24-Nov-2018) Thierry Arnoux created a new page on topological structures. The page along with its SVG files are maintained on GitHub.
(11-Oct-2018) Alexander van der Vekens added a new proof to the 100 theorem list, the Friendship Theorem friendship, bringing the Metamath total to 69.
(1-Oct-2018) Naip Moro has written gramm, a Metamath proof verifier written in Antlr4/Java.
(16-Sep-2018) The definition df-riota has been simplified so that it evaluates to the empty set instead of an Undef value. This change affects a significant part of set.mm.
(2-Sep-2018) Thierry Arnoux added a new proof to the 100 theorem list, Euler's partition theorem eulerpart, bringing the Metamath total to 68.
(1-Sep-2018) The Kate editor now has Metamath syntax highlighting built in. (Communicated by Wolf Lammen.)
(15-Aug-2018) The Intuitionistic Logic Explorer now has a Most Recent Proofs page.
(4-Aug-2018) Version 0.163 of the metamath program now indicates (with an asterisk) which Table of Contents headers have associated comments.
(10-May-2018) George Szpiro, journalist and author of several books on popular mathematics such as Poincare's Prize and Numbers Rule, used a genetic algorithm to find shorter D-proofs of "*3.37" and "meredith" in our Shortest known proofs... file.
(19-Apr-2018) The EMetamath Eclipse plugin has undergone many improvements since its initial release as the change log indicates. Thierry uses it as his main proof assistant and writes, "I added support for mmj2's auto-transformations, which allows it to infer several steps when building proofs. This added a lot of comfort for writing proofs.... I can now switch back and forth between the proof assistant and editing the Metamath file.... I think no other proof assistant has this feature."
(11-Apr-2018) Benoît Jubin solved an open problem about the "Axiom of Twoness," showing that it is necessary for completeness. See item 14 on the "Open problems and miscellany" page.
(25-Mar-2018) Giovanni Mascellani has announced mmpp, a new proof editing environment for the Metamath language.
(27-Feb-2018) Bill Hale has released an app for the Apple iPad and desktop computer that allows you to browse Metamath theorems and their proofs.
(17-Jan-2018) Dylan Houlihan has kindly provided a new mirror site. He has also provided an rsync server; type "rsync uk.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").
(15-Jan-2018) The metamath program, version 0.157, has been updated to implement the file inclusion conventions described in the 21-Dec-2017 entry of mmnotes.txt.
(11-Dec-2017) I added a paragraph, suggested by Gérard Lang, to the distinct variable description here.
(10-Dec-2017) Per FL's request, his mathbox will be removed from set.mm. If you wish to export any of his theorems, today's version (master commit 1024a3a) is the last one that will contain it.
(11-Nov-2017) Alan Sare updated his completeusersproof program.
(3-Oct-2017) Sean B. Palmer created a web page that runs the metamath program under emulated Linux in JavaScript. He also wrote some programs to work with our shortest known proofs of the PM propositional calculus theorems.
(28-Sep-2017) Ivan Kuckir wrote a tutorial blog entry, Introduction to Metamath, that summarizes the language syntax. (It may have been written some time ago, but I was not aware of it before.)
(26-Sep-2017) The default directory for the Metamath Proof Explorer (MPE) has been changed from the GIF version (mpegif) to the Unicode version (mpeuni) throughout the site. Please let me know if you find broken links or other issues.
(24-Sep-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Ceva's Theorem cevath, bringing the Metamath total to 67.
(3-Sep-2017) Brendan Leahy added a new proof to the 100 theorem list, Area of a Circle areacirc, bringing the Metamath total to 66.
(7-Aug-2017) Mario Carneiro added a new proof to the 100 theorem list, Principle of Inclusion/Exclusion incexc, bringing the Metamath total to 65.
(1-Jul-2017) Glauco Siliprandi added a new proof to the 100 theorem list, Stirling's Formula stirling, bringing the Metamath total to 64. Related theorems include 2 versions of Wallis' formula for π (wallispi and wallispi2).
(7-May-2017) Thierry Arnoux added a new proof to the 100 theorem list, Betrand's Ballot Problem ballotth, bringing the Metamath total to 63.
(20-Apr-2017) Glauco Siliprandi added a new proof in the supplementary list on the 100 theorem list, Stone-Weierstrass Theorem stowei.
(28-Feb-2017) David Moews added a new proof to the 100 theorem list, Product of Segments of Chords chordthm, bringing the Metamath total to 62.
(1-Jan-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Isosceles triangle theorem isosctr, bringing the Metamath total to 61.
(1-Jan-2017) Mario Carneiro added 2 new proofs to the 100 theorem list, L'Hôpital's Rule lhop and Taylor's Theorem taylth, bringing the Metamath total to 60.
(28-Dec-2016) David A. Wheeler is putting together a page on Metamath (specifically set.mm) conventions. Comments are welcome on the Google Group thread.
(24-Dec-2016) Mario Carneiro introduced the abbreviation "F/ x ph" (symbols: turned F, x, phi) in df-nf to represent the "effectively not free" idiom "A. x ( ph -> A. x ph )". Theorem nf2 shows a version without nested quantifiers.
(22-Dec-2016) Naip Moro has developed a Metamath database for G. Spencer-Brown's Laws of Form. You can follow the Google Group discussion here.
(20-Dec-2016) In metamath program version 0.137, 'verify markup *' now checks that ax-XXX $a matches axXXX $p when the latter exists, per the discussion at https://groups.google.com/d/msg/metamath/Vtz3CKGmXnI/Fxq3j1I_EQAJ.
(24-Nov-2016) Mingl Yuan has kindly provided a mirror site in Beijing, China. He has also provided an rsync server; type "rsync cn.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").
(14-Aug-2016) All HTML pages on this site should now be mobile-friendly and pass the Mobile-Friendly Test. If you find one that does not, let me know.
(14-Aug-2016) Daniel Whalen wrote a paper describing the use of using deep learning to prove 14% of test theorems taken from set.mm: Holophrasm: a neural Automated Theorem Prover for higher-order logic. The associated program is called Holophrasm.
(14-Aug-2016) David A. Wheeler created a video called Metamath Proof Explorer: A Modern Principia Mathematica
(12-Aug-2016) A Gitter chat room has been created for Metamath.
(9-Aug-2016) Mario Carneiro wrote a Metamath proof verifier in the Scala language as part of the ongoing Metamath -> MMT import project
(9-Aug-2016) David A. Wheeler created a GitHub project called metamath-test (last execution run) to check that different verifiers both pass good databases and detect errors in defective ones.
(4-Aug-2016) Mario gave two presentations at CICM 2016.
(17-Jul-2016) Thierry Arnoux has written EMetamath, a Metamath plugin for the Eclipse IDE.
(16-Jul-2016) Mario recovered Chris Capel's collapsible proof demo.
(13-Jul-2016) FL sent me an updated version of PDF (LaTeX source) developed with Lamport's pf2 package. See the 23-Apr-2012 entry below.
(12-Jul-2016) David A. Wheeler produced a new video for mmj2 called "Creating functions in Metamath". It shows a more efficient approach than his previous recent video "Creating functions in Metamath" (old) but it can be of interest to see both approaches.
(10-Jul-2016) Metamath program version 0.132 changes the command 'show restricted' to 'show discouraged' and adds a new command, 'set discouragement'. See the mmnotes.txt entry of 11-May-2016 (updated 10-Jul-2016).
(12-Jun-2016) Dan Getz has written Metamath.jl, a Metamath proof verifier written in the Julia language.
(10-Jun-2016) If you are using metamath program versions 0.128, 0.129, or 0.130, please update to version 0.131. (In the bad versions, 'minimize_with' ignores distinct variable violations.)
(1-Jun-2016) Mario Carneiro added new proofs to the 100 theorem list, the Prime Number Theorem pnt and the Perfect Number Theorem perfect, bringing the Metamath total to 58.
(12-May-2016) Mario Carneiro added a new proof to the 100 theorem list, Dirichlet's theorem dirith, bringing the Metamath total to 56. (Added 17-May-2016) An informal exposition of the proof can be found at http://metamath-blog.blogspot.com/2016/05/dirichlets-theorem.html
(10-Mar-2016) Metamath program version 0.125 adds a new qualifier, /fast, to 'save proof'. See the mmnotes.txt entry of 10-Mar-2016.
(6-Mar-2016) The most recent set.mm has a large update converting variables from letters to symbols. See this Google Groups post.
(16-Feb-2016) Mario Carneiro's new paper "Models for Metamath" can be found here and on arxiv.org.
(6-Feb-2016) There are now 22 math symbols that can be used as variable names. See mmascii.html near the 50th table row, starting with "./\".
(29-Jan-2016) Metamath program version 0.123 adds /packed and /explicit qualifiers to 'save proof' and 'show proof'. See this Google Groups post.
(13-Jan-2016) The Unicode math symbols now provide for external CSS and use the XITS web font. Thanks to David A. Wheeler, Mario Carneiro, Cris Perdue, Jason Orendorff, and Frédéric Liné for discussions on this topic. Two commands, htmlcss and htmlfont, were added to the $t comment in set.mm and are recognized by Metamath program version 0.122.
(21-Dec-2015) Axiom ax-12, now renamed ax-12o, was replaced by a new shorter equivalent, ax-12. The equivalence is provided by theorems ax12o and ax12.
(13-Dec-2015) A new section on the theory of classes was added to the MPE Home Page. Thanks to Gérard Lang for suggesting this section and improvements to it.
(17-Nov-2015) Metamath program version 0.121: 'verify markup' was added to check comment markup consistency; see 'help verify markup'. You are encouraged to make sure 'verify markup */f' has no warnings prior to mathbox submissions. The date consistency rules are given in this Google Groups post.
(23-Sep-2015) Drahflow wrote, "I am currently working on yet another proof assistant, main reason being: I understand stuff best if I code it. If anyone is interested: https://github.com/Drahflow/Igor (but in my own programming language, so expect a complicated build process :P)"
(23-Aug-2015) Ivan Kuckir created MM Tool, a Metamath proof verifier and editor written in JavaScript that runs in a browser.
(25-Jul-2015) Axiom ax-10 is shown to be redundant by theorem ax10 , so it was removed from the predicate calculus axiom list.
(19-Jul-2015) Mario Carneiro gave two talks related to Metamath at CICM 2015, which are linked to at Other Metamath-Related Topics.
(18-Jul-2015) The metamath program has been updated to version 0.118. 'show trace_back' now has a '/to' qualifier to show the path back to a specific axiom such as ax-ac. See 'help show trace_back'.
(12-Jul-2015) I added the HOL Explorer for Mario Carneiro's hol.mm database. Although the home page needs to be filled out, the proofs can be accessed.
(11-Jul-2015) I started a new page, Other Metamath-Related Topics, that will hold miscellaneous material that doesn't fit well elsewhere (or is hard to find on this site). Suggestions welcome.
(23-Jun-2015) Metamath's mascot, Penny the cat (2007 photo), passed away today. She was 18 years old.
(21-Jun-2015) Mario Carneiro added 3 new proofs to the 100 theorem list: All Primes (1 mod 4) Equal the Sum of Two Squares 2sq, The Law of Quadratic Reciprocity lgsquad and the AM-GM theorem amgm, bringing the Metamath total to 55.
(13-Jun-2015) Stefan O'Rear's smm, written in JavaScript, can now be used as a standalone proof verifier. This brings the total number of independent Metamath verifiers to 8, written in just as many languages (C, Java. JavaScript, Python, Haskell, Lua, C#, C++).
(12-Jun-2015) David A. Wheeler added 2 new proofs to the 100 theorem list: The Law of Cosines lawcos and Ptolemy's Theorem ptolemy, bringing the Metamath total to 52.
(30-May-2015) The metamath program has been updated to version 0.117. (1) David A. Wheeler provided an enhancement to speed up the 'improve' command by 28%; see README.TXT for more information. (2) In web pages with proofs, local hyperlinks on step hypotheses no longer clip the Expression cell at the top of the page.
(9-May-2015) Stefan O'Rear has created an archive of older set.mm releases back to 1998: https://github.com/sorear/set.mm-history/.
(7-May-2015) The set.mm dated 7-May-2015 is a major revision, updated by Mario, that incorporates the new ordered pair definition df-op that was agreed upon. There were 700 changes, listed at the top of set.mm. Mathbox users are advised to update their local mathboxes. As usual, if any mathbox user has trouble incorporating these changes into their mathbox in progress, Mario or I will be glad to do them for you.
(7-May-2015) Mario has added 4 new theorems to the 100 theorem list: Ramsey's Theorem ramsey, The Solution of a Cubic cubic, The Solution of the General Quartic Equation quart, and The Birthday Problem birthday. In the Supplementary List, Stefan O'Rear added the Hilbert Basis Theorem hbt.
(28-Apr-2015) A while ago, Mario Carneiro wrote up a proof of the unambiguity of set.mm's grammar, which has now been added to this site: grammar-ambiguity.txt.
(22-Apr-2015) The metamath program has been updated to version 0.114. In MM-PA, 'show new_proof/unknown' now shows the relative offset (-1, -2,...) used for 'assign' arguments, suggested by Stefan O'Rear.
(20-Apr-2015) I retrieved an old version of the missing "Metamath 100" page from archive.org and updated it to what I think is the current state: mm_100.html. Anyone who wants to edit it can email updates to this page to me.
(19-Apr-2015) The metamath program has been updated to version 0.113, mostly with patches provided by Stefan O'Rear. (1) 'show statement %' (or any command allowing label wildcards) will select statements whose proofs were changed in current session. ('help search' will show all wildcard matching rules.) (2) 'show statement =' will select the statement being proved in MM-PA. (3) The proof date stamp is now created only if the proof is complete.
(18-Apr-2015) There is now a section for Scott Fenton's NF database: New Foundations Explorer.
(16-Apr-2015) Mario describes his recent additions to set.mm at https://groups.google.com/forum/#!topic/metamath/VAGNmzFkHCs. It include 2 new additions to the Formalizing 100 Theorems list, Leibniz' series for pi (leibpi) and the Konigsberg Bridge problem (konigsberg)
(10-Mar-2015) Mario Carneiro has written a paper, "Arithmetic in Metamath, Case Study: Bertrand's Postulate," for CICM 2015. A preprint is available at arXiv:1503.02349.
(23-Feb-2015) Scott Fenton has created a Metamath formalization of NF set theory: https://github.com/sctfn/metamath-nf/. For more information, see the Metamath Google Group posting.
(28-Jan-2015) Mario Carneiro added Wilson's Theorem (wilth), Ascending or Descending Sequences (erdsze, erdsze2), and Derangements Formula (derangfmla, subfaclim), bringing the Metamath total for Formalizing 100 Theorems to 44.
(19-Jan-2015) Mario Carneiro added Sylow's Theorem (sylow1, sylow2, sylow2b, sylow3), bringing the Metamath total for Formalizing 100 Theorems to 41.
(9-Jan-2015) The hypothesis order of mpbi*an* was changed. See the Notes entry of 9-Jan-2015.
(1-Jan-2015) Mario Carneiro has written a paper, "Conversion of HOL Light proofs into Metamath," that has been submitted to the Journal of Formalized Reasoning. A preprint is available on arxiv.org.
(22-Nov-2014) Stefan O'Rear added the Solutions to Pell's Equation (rmxycomplete) and Liouville's Theorem and the Construction of Transcendental Numbers (aaliou), bringing the Metamath total for Formalizing 100 Theorems to 40.
(22-Nov-2014) The metamath program has been updated with version 0.111. (1) Label wildcards now have a label range indicator "~" so that e.g. you can show or search all of the statements in a mathbox. See 'help search'. (Stefan O'Rear added this to the program.) (2) A qualifier was added to 'minimize_with' to prevent the use of any axioms not already used in the proof e.g. 'minimize_with * /no_new_axioms_from ax-*' will prevent the use of ax-ac if the proof doesn't already use it. See 'help minimize_with'.
(10-Oct-2014) Mario Carneiro has encoded the axiomatic basis for the HOL theorem prover into a Metamath source file, hol.mm.
(24-Sep-2014) Mario Carneiro added the Sum of the Angles of a Triangle (ang180), bringing the Metamath total for Formalizing 100 Theorems to 38.
(15-Sep-2014) Mario Carneiro added the Fundamental Theorem of Algebra (fta), bringing the Metamath total for Formalizing 100 Theorems to 37.
(3-Sep-2014) Mario Carneiro added the Fundamental Theorem of Integral Calculus (ftc1, ftc2). This brings the Metamath total for Formalizing 100 Theorems to 35. (added 14-Sep-2014) Along the way, he added the Mean Value Theorem (mvth), bringing the total to 36.
(16-Aug-2014) Mario Carneiro started a Metamath blog at http://metamath-blog.blogspot.com/.
(10-Aug-2014) Mario Carneiro added Erdős's proof of the divergence of the inverse prime series (prmrec). This brings the Metamath total for Formalizing 100 Theorems to 34.
(31-Jul-2014) Mario Carneiro added proofs for Euler's Summation of 1 + (1/2)^2 + (1/3)^2 + .... (basel) and The Factor and Remainder Theorems (facth, plyrem). This brings the Metamath total for Formalizing 100 Theorems to 33.
(16-Jul-2014) Mario Carneiro added proofs for Four Squares Theorem (4sq), Formula for the Number of Combinations (hashbc), and Divisibility by 3 Rule (3dvds). This brings the Metamath total for Formalizing 100 Theorems to 31.
(11-Jul-2014) Mario Carneiro added proofs for Divergence of the Harmonic Series (harmonic), Order of a Subgroup (lagsubg), and Lebesgue Measure and Integration (itgcl). This brings the Metamath total for Formalizing 100 Theorems to 28.
(7-Jul-2014) Mario Carneiro presented a talk, "Natural Deduction in the Metamath Proof Language," at the 6PCM conference. Slides Audio
(25-Jun-2014) In version 0.108 of the metamath program, the 'minimize_with' command is now more automated. It now considers compressed proof length; it scans the statements in forward and reverse order and chooses the best; and it avoids $d conflicts. The '/no_distinct', '/brief', and '/reverse' qualifiers are obsolete, and '/verbose' no longer lists all statements scanned but gives more details about decision criteria.
(12-Jun-2014) To improve naming uniformity, theorems about operation values now use the abbreviation "ov". For example, df-opr, opreq1, oprabval5, and oprvres are now called df-ov, oveq1, ov5, and ovres respectively.
(11-Jun-2014) Mario Carneiro finished a major revision of set.mm. His notes are under the 11-Jun-2014 entry in the Notes
(4-Jun-2014) Mario Carneiro provided instructions and screenshots for syntax highlighting for the jEdit editor for use with Metamath and mmj2 source files.
(19-May-2014) Mario Carneiro added a feature to mmj2, in the build at
https://github.com/digama0/mmj2/raw/dev-build/mmj2jar/mmj2.jar, which
tests all but 5 definitions in set.mm for soundness. You can turn on
the test by adding
SetMMDefinitionsCheckWithExclusions,ax-*,df-bi,df-clab,df-cleq,df-clel,df-sbc
to your RunParms.txt file.
(17-May-2014) A number of labels were changed in set.mm, listed at the top of set.mm as usual. Note in particular that the heavily-used visset, elisseti, syl11anc, syl111anc were changed respectively to vex, elexi, syl2anc, syl3anc.
(16-May-2014) Scott Fenton formalized a proof for "Sum of kth powers": fsumkthpow. This brings the Metamath total for Formalizing 100 Theorems to 25.
(9-May-2014) I (Norm Megill) presented an overview of Metamath at the "Formalization of mathematics in proof assistants" workshop at the Institut Henri Poincaré in Paris. The slides for this talk are here.
(22-Jun-2014) Version 0.107 of the metamath program adds a "PART" indention level to the Statement List table of contents, adds 'show proof ... /size' to show source file bytes used, and adds 'show elapsed_time'. The last one is helpful for measuring the run time of long commands. See 'help write theorem_list', 'help show proof', and 'help show elapsed_time' for more information.
(2-May-2014) Scott Fenton formalized a proof of Sum of the Reciprocals of the Triangular Numbers: trirecip. This brings the Metamath total for Formalizing 100 Theorems to 24.
(19-Apr-2014) Scott Fenton formalized a proof of the Formula for Pythagorean Triples: pythagtrip. This brings the Metamath total for Formalizing 100 Theorems to 23.
(11-Apr-2014) David A. Wheeler produced a much-needed and well-done video for mmj2, called "Introduction to Metamath & mmj2". Thanks, David!
(15-Mar-2014) Mario Carneiro formalized a proof of Bertrand's postulate: bpos. This brings the Metamath total for Formalizing 100 Theorems to 22.
(18-Feb-2014) Mario Carneiro proved that complex number axiom ax-cnex is redundant (theorem cnex). See also Real and Complex Numbers.
(11-Feb-2014) David A. Wheeler has created a theorem compilation that tracks those theorems in Freek Wiedijk's Formalizing 100 Theorems list that have been proved in set.mm. If you find a error or omission in this list, let me know so it can be corrected. (Update 1-Mar-2014: Mario has added eulerth and bezout to the list.)
(4-Feb-2014) Mario Carneiro writes:
The latest commit on the mmj2 development branch introduced an exciting new feature, namely syntax highlighting for mmp files in the main window. (You can pick up the latest mmj2.jar at https://github.com/digama0/mmj2/blob/develop/mmj2jar/mmj2.jar .) The reason I am asking for your help at this stage is to help with design for the syntax tokenizer, which is responsible for breaking down the input into various tokens with names like "comment", "set", and "stephypref", which are then colored according to the user's preference. As users of mmj2 and metamath, what types of highlighting would be useful to you?One limitation of the tokenizer is that since (for performance reasons) it can be started at any line in the file, highly contextual coloring, like highlighting step references that don't exist previously in the file, is difficult to do. Similarly, true parsing of the formulas using the grammar is possible but likely to be unmanageably slow. But things like checking theorem labels against the database is quite simple to do under the current setup.
That said, how can this new feature be optimized to help you when writing proofs?
(13-Jan-2014) Mathbox users: the *19.21a*, *19.23a* series of theorems have been renamed to *alrim*, *exlim*. You can update your mathbox with a global replacement of string '19.21a' with 'alrim' and '19.23a' with 'exlim'.
(5-Jan-2014) If you downloaded mmj2 in the past 3 days, please update it with the current version, which fixes a bug introduced by the recent changes that made it unable to read in most of the proofs in the textarea properly.
(4-Jan-2014) I added a list of "Allowed substitutions" under the "Distinct variable groups" list on the theorem web pages, for example axsep. This is an experimental feature and comments are welcome.
(3-Jan-2014) Version 0.102 of the metamath program produces more space-efficient compressed proofs (still compatible with the specification in Appendix B of the Metamath book) using an algorithm suggested by Mario Carneiro. See 'help save proof' in the program. Also, mmj2 now generates proofs in the new format. The new mmj2 also has a mandatory update that fixes a bug related to the new format; you must update your mmj2 copy to use it with the latest set.mm.
(23-Dec-2013) Mario Carneiro has updated many older definitions to use the maps-to notation. If you have difficulty updating your local mathbox, contact him or me for assistance.
(1-Nov-2013) 'undo' and 'redo' commands were added to the Proof Assistant in metamath program version 0.07.99. See 'help undo' in the program.
(8-Oct-2013) Today's Notes entry describes some proof repair techniques.
(5-Oct-2013) Today's Notes entry explains some recent extensible structure improvements.
(8-Sep-2013) Mario Carneiro has revised the square root and sequence generator definitions. See today's Notes entry.
(3-Aug-2013) Mario Carneiro writes: "I finally found enough time to create a GitHub repository for development at https://github.com/digama0/mmj2. A permalink to the latest version plus source (akin to mmj2.zip) is https://github.com/digama0/mmj2/zipball/, and the jar file on its own (mmj2.jar) is at https://github.com/digama0/mmj2/blob/master/mmj2jar/mmj2.jar?raw=true. Unfortunately there is no easy way to automatically generate mmj2jar.zip, but this is available as part of the zip distribution for mmj2.zip. History tracking will be handled by the repository now. Do you have old versions of the mmj2 directory? I could add them as historical commits if you do."
(18-Jun-2013) Mario Carneiro has done a major revision and cleanup of the construction of real and complex numbers. In particular, rather than using equivalence classes as is customary for the construction of the temporary rationals, he used only "reduced fractions", so that the use of the axiom of infinity is avoided until it becomes necessary for the construction of the temporary reals.
(18-May-2013) Mario Carneiro has added the ability to produce compressed proofs to mmj2. This is not an official release but can be downloaded here if you want to try it: mmj2.jar. If you have any feedback, send it to me (NM), and I will forward it to Mario. (Disclaimer: this release has not been endorsed by Mel O'Cat. If anyone has been in contact with him, please let me know.)
(29-Mar-2013) Charles Greathouse reduced the size of our PNG symbol images using the pngout program.
(8-Mar-2013) Wolf Lammen has reorganized the theorems in the "Logical negation" section of set.mm into a more orderly, less scattered arrangement.
(27-Feb-2013) Scott Fenton has done a large cleanup of set.mm, eliminating *OLD references in 144 proofs. See the Notes entry for 27-Feb-2013.
(21-Feb-2013) *ATTENTION MATHBOX USERS* The order of hypotheses of many syl* theorems were changed, per a suggestion of Mario Carneiro. You need to update your local mathbox copy for compatibility with the new set.mm, or I can do it for you if you wish. See the Notes entry for 21-Feb-2013.
(16-Feb-2013) Scott Fenton shortened the direct-from-axiom proofs of *3.1, *3.43, *4.4, *4.41, *4.5, *4.76, *4.83, *5.33, *5.35, *5.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).
(27-Jan-2013) Scott Fenton writes, "I've updated Ralph Levien's mmverify.py. It's now a Python 3 program, and supports compressed proofs and file inclusion statements. This adds about fifty lines to the original program. Enjoy!"
(10-Jan-2013) A new mathbox was added for Mario Carneiro, who has contributed a number of cardinality theorems without invoking the Axiom of Choice. This is nice work, and I will be using some of these (those suffixed with "NEW") to replace the existing ones in the main part of set.mm that currently invoke AC unnecessarily.
(4-Jan-2013) As mentioned in the 19-Jun-2012 item below, Eric Schmidt discovered that the complex number axioms axaddcom (now addcom) and ax0id (now addid1) are redundant (schmidt-cnaxioms.pdf, .tex). In addition, ax1id (now mulid1) can be weakened to ax1rid. Scott Fenton has now formalized this work, so that now there are 23 instead of 25 axioms for real and complex numbers in set.mm. The Axioms for Complex Numbers page has been updated with these results. An interesting part of the proof, showing how commutativity of addition follows from other laws, is in addcomi.
(27-Nov-2012) The frequently-used theorems "an1s", "an1rs", "ancom13s", "ancom31s" were renamed to "an12s", "an32s", "an13s", "an31s" to conform to the convention for an12 etc.
(4-Nov-2012) The changes proposed in the Notes, renaming Grp to GrpOp etc., have been incorporated into set.mm. See the list of changes at the top of set.mm. If you want me to update your mathbox with these changes, send it to me along with the version of set.mm that it works with.
(20-Sep-2012) Mel O'Cat updated http://us2.metamath.org:88/ocat/mmj2/TESTmmj2jar.zip. See the README.TXT for a description of the new features.
(21-Aug-2012) Mel O'Cat has uploaded SearchOptionsMockup9.zip, a mockup for the new search screen in mmj2. See the README.txt file for instructions. He will welcome feedback via x178g243 at yahoo.com.
(19-Jun-2012) Eric Schmidt has discovered that in our axioms for complex numbers, axaddcom and ax0id are redundant. (At some point these need to be formalized for set.mm.) He has written up these and some other nice results, including some independence results for the axioms, in schmidt-cnaxioms.pdf (schmidt-cnaxioms.tex).
(23-Apr-2012) Frédéric Liné sent me a PDF (LaTeX source) developed with Lamport's pf2 package. He wrote: "I think it works well with Metamath since the proofs are in a tree form. I use it to have a sketch of a proof. I get this way a better understanding of the proof and I can cut down its size. For instance, inpreima5 was reduced by 50% when I wrote the corresponding proof with pf2."
(5-Mar-2012) I added links to Wikiproofs and its recent changes in the "Wikis" list at the top of this page.
(12-Jan-2012) Thanks to William Hoza who sent me a ZFC T-shirt, and thanks to the ZFC models (courtesy of the Inaccessible Cardinals agency).
Front | Back | Detail |
(24-Nov-2011) In metamath program version 0.07.71, the 'minimize_with' command by default now scans from bottom to top instead of top to bottom, since empirically this often (although not always) results in a shorter proof. A top to bottom scan can be specified with a new qualifier '/reverse'. You can try both methods (starting from the same original proof, of course) and pick the shorter proof.
(15-Oct-2011) From Mel O'Cat:
I just uploaded mmj2.zip containing the 1-Nov-2011 (20111101)
release:
http://us2.metamath.org:88/ocat/mmj2/mmj2.zip
http://us2.metamath.org:88/ocat/mmj2/mmj2.md5
A few last minute tweaks:
1. I now bless double-click starting of mmj2.bat (MacMMJ2.command in Mac OS-X)!
See mmj2\QuickStart.html
2. Much improved support of Mac OS-X systems.
See mmj2\QuickStart.html
3. I tweaked the Command Line Argument Options report to
a) print every time;
b) print as much as possible even if
there are errors in the command line arguments -- and the
last line printed corresponds to the argument in error;
c) removed Y/N argument on the command line to enable/disable
the report. this simplifies things.
4) Documentation revised, including the PATutorial.
See CHGLOG.TXT for list of all changes.
Good luck. And thanks for all of your help!
(15-Sep-2011) MATHBOX USERS: I made a large number of label name changes to set.mm to improve naming consistency. There is a script at the top of the current set.mm that you can use to update your mathbox or older set.mm. Or if you wish, I can do the update on your next mathbox submission - in that case, please include a .zip of the set.mm version you used.
(30-Aug-2011) Scott Fenton shortened the direct-from-axiom proofs of *3.33, *3.45, *4.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).
(21-Aug-2011) A post on reddit generated 60,000 hits (and a TOS violation notice from my provider...),
(18-Aug-2011) The Metamath Google Group has a discussion of my canonical conjunctions proposal. Any feedback directly to me (Norm Megill) is also welcome.
(4-Jul-2011) John Baker has provided (metamath_kindle.zip) "a modified version of [the] metamath.tex [Metamath] book source that is formatted for the Kindle. If you compile the document the resulting PDF can be loaded into into a Kindle and easily read." (Update: the PDF file is now included also.)
(3-Jul-2011) Nested 'submit' calls are now allowed, in metamath program version 0.07.68. Thus you can create or modify a command file (script) from within a command file then 'submit' it. While 'submit' cannot pass arguments (nor are there plans to add this feature), you can 'substitute' strings in the 'submit' target file before calling it in order to emulate this.
(28-Jun-2011)The metamath program version 0.07.64 adds the '/include_mathboxes' qualifier to 'minimize_with'; by default, 'minimize_with *' will now skip checking user mathboxes. Since mathboxes should be independent from each other, this will help prevent accidental cross-"contamination". Also, '/rewrap' was added to 'write source' to automatically wrap $a and $p comments so as to conform to the current formatting conventions used in set.mm. This means you no longer have to be concerned about line length < 80 etc.
(19-Jun-2011) ATTENTION MATHBOX USERS: The wff variables et, ze, si, and rh are now global. This change was made primarily to resolve some conflicts between mathboxes, but it will also let you avoid having to constantly redeclare these locally in the future. Unfortunately, this change can affect the $f hypothesis order, which can cause proofs referencing theorems that use these variables to fail. All mathbox proofs currently in set.mm have been corrected for this, and you should refresh your local copy for further development of your mathbox. You can correct your proofs that are not in set.mm as follows. Only the proofs that fail under the current set.mm (using version 0.07.62 or later of the metamath program) need to be modified.
To fix a proof that references earlier theorems using et, ze, si, and rh, do the following (using a hypothetical theorem 'abc' as an example): 'prove abc' (ignore error messages), 'delete floating', 'initialize all', 'unify all/interactive', 'improve all', 'save new_proof/compressed'. If your proof uses dummy variables, these must be reassigned manually.
To fix a proof that uses et, ze, si, and rh as local variables, make sure the proof is saved in 'compressed' format. Then delete the local declarations ($v and $f statements) and follow the same steps above to correct the proof.
I apologize for the inconvenience. If you have trouble fixing your proofs, you can contact me for assistance.
Note: Versions of the metamath program before 0.07.62 did not flag an error when global variables were redeclared locally, as it should have according to the spec. This caused these spec violations to go unnoticed in some older set.mm versions. The new error messages are in fact just informational and can be ignored when working with older set.mm versions.
(7-Jun-2011) The metamath program version 0.07.60 fixes a bug with the 'minimize_with' command found by Andrew Salmon.
(12-May-2010) Andrew Salmon shortened many proofs, shown above. For comparison, I have temporarily kept the old version, which is suffixed with OLD, such as oridmOLD for oridm.
(9-Dec-2010) Eric Schmidt has written a Metamath proof verifier in C++, called checkmm.cpp.
(3-Oct-2010) The following changes were made to the tokens in set.mm. The subset and proper subset symbol changes to C_ and C. were made to prevent defeating the parenthesis matching in Emacs. Other changes were made so that all letters a-z and A-Z are now available for variable names. One-letter constants such as _V, _e, and _i are now shown on the web pages with Roman instead of italic font, to disambiguate italic variable names. The new convention is that a prefix of _ indicates Roman font and a prefix of ~ indicates a script (curly) font. Thanks to Stefan Allan and Frédéric Liné for discussions leading to this change.
Old | New | Description |
---|---|---|
C. | _C | binomial coefficient |
E | _E | epsilon relation |
e | _e | Euler's constant |
I | _I | identity relation |
i | _i | imaginary unit |
V | _V | universal class |
(_ | C_ | subset |
(. | C. | proper subset |
P~ | ~P | power class |
H~ | ~H | Hilbert space |
(25-Sep-2010) The metamath program (version 0.07.54) now implements the current Metamath spec, so footnote 2 on p. 92 of the Metamath book can be ignored.
(24-Sep-2010) The metamath program (version 0.07.53) fixes bug 2106, reported by Michal Burger.
(14-Sep-2010) The metamath program (version 0.07.52) has a revamped LaTeX output with 'show statement xxx /tex', which produces the combined statement, description, and proof similar to the web page generation. Also, 'show proof xxx /lemmon/renumber' now matches the web page step numbers. ('show proof xxx/renumber' still has the indented form conforming to the actual RPN proof, with slightly different numbering.)
(9-Sep-2010) The metamath program (version 0.07.51) was updated with a modification by Stefan Allan that adds hyperlinks the the Ref column of proofs.
(12-Jun-2010) Scott Fenton contributed a D-proof (directly from axioms) of Meredith's single axiom (see the end of pmproofs.txt). A description of Meredith's axiom can be found in theorem meredith.
(11-Jun-2010) A new Metamath mirror was added in Austria, courtesy of Kinder-Enduro.
(28-Feb-2010) Raph Levien's Ghilbert project now has a new Ghilbert site and a Google Group.
(26-Jan-2010) Dmitri Vlasov writes, "I admire the simplicity and power of the metamath language, but still I see its great disadvantage - the proofs in metamath are completely non-manageable by humans without proof assistants. Therefore I decided to develop another language, which would be a higher-level superstructure language towards metamath, and which will support human-readable/writable proofs directly, without proof assistants. I call this language mdl (acronym for 'mathematics development language')." The latest version of Dmitri's translators from metamath to mdl and back can be downloaded from http://mathdevlanguage.sourceforge.net/. Currently only Linux is supported, but Dmitri says is should not be difficult to port it to other platforms that have a g++ compiler.
(11-Sep-2009) The metamath program (version 0.07.48) has been updated to enforce the whitespace requirement of the current spec.
(10-Sep-2009) Matthew Leitch has written an nice article, "How to write mathematics clearly", that briefly mentions Metamath. Overall it makes some excellent points. (I have written to him about a few things I disagree with.)
(28-May-2009) AsteroidMeta is back on-line. Note the URL change.
(12-May-2009) Charles Greathouse wrote a Greasemonkey script to reformat the axiom list on Metamath web site proof pages. This is a beta version; he will appreciate feedback.
(11-May-2009) Stefan Allan modified the metamath program to add the command "show statement xxx /mnemonics", which produces the output file Mnemosyne.txt for use with the Mnemosyne project. The current Metamath program download incorporates this command. Instructions: Create the file mnemosyne.txt with e.g. "show statement ax-* /mnemonics". In the Mnemosyne program, load the file by choosing File->Import then file format "Q and A on separate lines". Notes: (1) Don't try to load all of set.mm, it will crash the program due to a bug in Mnemosyne. (2) On my computer, the arrows in ax-1 don't display. Stefan reports that they do on his computer. (Both are Windows XP.)
(3-May-2009) Steven Baldasty wrote a Metamath syntax highlighting file for the gedit editor. Screenshot.
(1-May-2009) Users on a gaming forum discuss our 2+2=4 proof. Notable comments include "Ew math!" and "Whoever wrote this has absolutely no life."
(12-Mar-2009) Chris Capel has created a Javascript theorem viewer demo that (1) shows substitutions and (2) allows expanding and collapsing proof steps. You are invited to take a look and give him feedback at his Metablog.
(28-Feb-2009) Chris Capel has written a Metamath proof verifier in C#, available at http://pdf23ds.net/bzr/MathEditor/Verifier/Verifier.cs and weighing in at 550 lines. Also, that same URL without the file on it is a Bazaar repository.
(2-Dec-2008) A new section was added to the Deduction Theorem page, called Logic, Metalogic, Metametalogic, and Metametametalogic.
(24-Aug-2008) (From ocat): The 1-Aug-2008 version of mmj2 is ready (mmj2.zip), size = 1,534,041 bytes. This version contains the Theorem Loader enhancement which provides a "sandboxing" capability for user theorems and dynamic update of new theorems to the Metamath database already loaded in memory by mmj2. Also, the new "mmj2 Service" feature enables calling mmj2 as a subroutine, or having mmj2 call your program, and provides access to the mmj2 data structures and objects loaded in memory (i.e. get started writing those Jython programs!) See also mmj2 on AsteroidMeta.
(23-May-2008) Gérard Lang pointed me to Bob Solovay's note on AC and strongly inaccessible cardinals. One of the eventual goals for set.mm is to prove the Axiom of Choice from Grothendieck's axiom, like Mizar does, and this note may be helpful for anyone wanting to attempt that. Separately, I also came across a history of the size reduction of grothprim (viewable in Firefox and some versions of Internet Explorer).
(14-Apr-2008) A "/join" qualifier was added to the "search" command in the metamath program (version 0.07.37). This qualifier will join the $e hypotheses to the $a or $p for searching, so that math tokens in the $e's can be matched as well. For example, "search *com* +v" produces no results, but "search *com* +v /join" yields commutative laws involving vector addition. Thanks to Stefan Allan for suggesting this idea.
(8-Apr-2008) The 8,000th theorem, hlrel, was added to the Metamath Proof Explorer part of the database.
(2-Mar-2008) I added a small section to the end of the Deduction Theorem page.
(17-Feb-2008) ocat has uploaded the "1-Mar-2008" mmj2: mmj2.zip. See the description.
(16-Jan-2008) O'Cat has written mmj2 Proof Assistant Quick Tips.
(30-Dec-2007) "How to build a library of formalized mathematics".
(22-Dec-2007) The Metamath Proof Explorer was included in the top 30 science resources for 2007 by the University at Albany Science Library.
(17-Dec-2007) Metamath's Wikipedia entry says, "This article may require cleanup to meet Wikipedia's quality standards" (see its discussion page). Volunteers are welcome. :) (In the interest of objectivity, I don't edit this entry.)
(20-Nov-2007) Jeff Hoffman created nicod.mm and posted it to the Google Metamath Group.
(19-Nov-2007) Reinder Verlinde suggested adding tooltips to the hyperlinks on the proof pages, which I did for proof step hyperlinks. Discussion.
(5-Nov-2007) A Usenet challenge. :)
(4-Aug-2007) I added a "Request for comments on proposed 'maps to' notation" at the bottom of the AsteroidMeta set.mm discussion page.
(21-Jun-2007) A preprint (PDF file) describing Kurt Maes' axiom of choice with 5 quantifiers, proved in set.mm as ackm.
(20-Jun-2007) The 7,000th theorem, ifpr, was added to the Metamath Proof Explorer part of the database.
(29-Apr-2007) Blog mentions of Metamath: here and here.
(21-Mar-2007) Paul Chapman is working on a new proof browser, which has highlighting that allows you to see the referenced theorem before and after the substitution was made. Here is a screenshot of theorem 0nn0 and a screenshot of theorem 2p2e4.
(15-Mar-2007) A picture of Penny the cat guarding the us2.metamath.org:8888 server and making the rounds.
(16-Feb-2007) For convenience, the program "drule.c" (pronounced "D-rule", not "drool") mentioned in pmproofs.txt can now be downloaded (drule.c) without having to ask me for it. The same disclaimer applies: even though this program works and has no known bugs, it was not intended for general release. Read the comments at the top of the program for instructions.
(28-Jan-2007) Jason Orendorff set up a new mailing list for Metamath: http://groups.google.com/group/metamath.
(20-Jan-2007) Bob Solovay provided a revised version of his Metamath database for Peano arithmetic, peano.mm.
(2-Jan-2007) Raph Levien has set up a wiki called Barghest for the Ghilbert language and software.
(26-Dec-2006) I posted an explanation of theorem ecoprass on Usenet.
(2-Dec-2006) Berislav Žarnić translated the Metamath Solitaire applet to Croatian.
(26-Nov-2006) Dan Getz has created an RSS feed for new theorems as they appear on this page.
(6-Nov-2006) The first 3 paragraphs in Appendix 2: Note on the Axioms were rewritten to clarify the connection between Tarski's axiom system and Metamath.
(31-Oct-2006) ocat asked for a do-over due to a bug in mmj2 -- if you downloaded the mmj2.zip version dated 10/28/2006, then download the new version dated 10/30.
(29-Oct-2006) ocat has announced that the
long-awaited 1-Nov-2006 release of mmj2 is available now.
The new "Unify+Get Hints" is quite
useful, and any proof can be generated as follows. With "?" in the Hyp
field and Ref field blank, select "Unify+Get Hints". Select a hint from
the list and put it in the Ref field. Edit any $n dummy variables to
become the desired wffs. Rinse and repeat for the new proof steps
generated, until the proof is done.
The new tutorial, mmj2PATutorial.bat,
explains this in detail. One way to reduce or avoid dummy $n's is to
fill in the Hyp field with a comma-separated list of any known
hypothesis matches to earlier proof steps, keeping a "?" in the list to
indicate that the remaining hypotheses are unknown. Then "Unify+Get
Hints" can be applied. The tutorial page
\mmj2\data\mmp\PATutorial\Page405.mmp has an example.
Don't forget that the eimm
export/import program lets you go back and forth between the mmj2 and
the metamath program proof assistants, without exiting from either one,
to exploit the best features of each as required.
(21-Oct-2006) Martin Kiselkov has written a Metamath proof verifier in the Lua scripting language, called verify.lua. While it is not practical as an everyday verifier - he writes that it takes about 40 minutes to verify set.mm on a a Pentium 4 - it could be useful to someone learning Lua or Metamath, and importantly it provides another independent way of verifying the correctness of Metamath proofs. His code looks like it is nicely structured and very readable. He is currently working on a faster version in C++.
(19-Oct-2006) New AsteroidMeta page by Raph, Distinctors_vs_binders.
(13-Oct-2006) I put a simple Metamath browser on my PDA (Palm Tungsten E) so that I don't have to lug around my laptop. Here is a screenshot. It isn't polished, but I'll provide the file + instructions if anyone wants it.
(3-Oct-2006) A blog entry, Principia for Reverse Mathematics.
(28-Sep-2006) A blog entry, Metamath responds.
(26-Sep-2006) A blog entry, Metamath isn't hygienic.
(11-Aug-2006) A blog entry, Metamath and the Peano Induction Axiom.
(26-Jul-2006) A new open problem in predicate calculus was added.
(18-Jun-2006) The 6,000th theorem, mt4d, was added to the Metamath Proof Explorer part of the database.
(9-May-2006) Luca Ciciriello has upgraded the t2mf program, which is a C
program used to create the MIDI files on the
Metamath Music Page, so
that it works on MacOS X. This is a nice accomplishment, since the
original program was written before C was standardized by ANSI and will
not compile on modern compilers.
Unfortunately, the original program source states no copyright terms.
The main author, Tim Thompson, has kindly agreed to release his code to
public domain, but two other authors have also contributed to the code,
and so far I have been unable to contact them for copyright clearance.
Therefore I cannot offer the MacOS X version for public download on this
site until this is resolved. Update 10-May-2006: Another author,
M. Czeiszperger, has released his contribution to public domain.
If you are interested in Luca's modified source code,
please contact me directly.
(18-Apr-2006) Incomplete proofs in progress can now be interchanged between the Metamath program's CLI Proof Assistant and mmj2's GUI Proof Assistant, using a new export-import program called eimm. This can be done without exiting either proof assistant, so that the strengths of each approach can be exploited during proof development. See "Use Case 5a" and "Use Case 5b" at mmj2ProofAssistantFeedback.
(28-Mar-2006) Scott Fenton updated his second version of Metamath Solitaire (the one that uses external axioms). He writes: "I've switched to making it a standalone program, as it seems silly to have an applet that can't be run in a web browser. Check the README file for further info." The download is mmsol-0.5.tar.gz.
(27-Mar-2006) Scott Fenton has updated the Metamath Solitaire Java
applet to Java 1.5: (1) QSort has been stripped out: its functionality
is in the Collections class that Sun ships; (2) all Vectors have been
replaced by ArrayLists; (3) generic types have been tossed in wherever
they fit: this cuts back drastically on casting; and (4) any warnings
Eclipse spouted out have been dealt with. I haven't yet updated it
officially, because I don't know if it will work with Microsoft's JVM in
older versions of Internet Explorer. The current official version is
compiled with Java 1.3, because it won't work with Microsoft's JVM if it
is compiled with Java 1.4. (As distasteful as that seems,
I will get complaints from users if it
doesn't work with Microsoft's JVM.) If anyone can verify that Scott's new
version runs on Microsoft's JVM, I would be grateful. Scott's new
version is mm.java-1.5.gz; after
uncompressing it, rename it to mm.java,
use it to replace the existing mm.java file in the
Metamath Solitaire download, and recompile according to instructions
in the mm.java comments.
Scott has also created a second version, mmsol-0.2.tar.gz, that reads
the axioms from ASCII files, instead of having the axioms hard-coded in
the program. This can be very useful if you want to play with custom
axioms, and you can also add a collection of starting theorems as
"axioms" to work from. However, it must be run from the local directory
with appletviewer, since the default Java security model doesn't allow
reading files from a browser. It works with the JDK 5 Update 6
Java download.
To compile (from Windows Command Prompt): C:\Program
Files\Java\jdk1.5.0_06\bin\javac.exe mm.java
To run (from Windows Command Prompt): C:\Program
Files\Java\jdk1.5.0_06\bin\appletviewer.exe mms.html
(21-Jan-2006) Juha Arpiainen proved the independence of axiom ax-11 from the others. This was published as an open problem in my 1995 paper (Remark 9.5 on PDF page 17). See Item 9a on the Workshop Miscellany for his seven-line proof. See also the Asteroid Meta metamathMathQuestions page under the heading "Axiom of variable substitution: ax-11". Congratulations, Juha!
(20-Oct-2005) Juha Arpiainen is working on a proof verifier in Common Lisp called Bourbaki. Its proof language has its roots in Metamath, with the goal of providing a more powerful syntax and definitional soundness checking. See its documentation and related discussion.
(17-Oct-2005) Marnix Klooster has written a Metamath proof verifier in Haskell, called Hmm. Also see his Announcement. The complete program (Hmm.hs, HmmImpl.hs, and HmmVerify.hs) has only 444 lines of code, excluding comments and blank lines. It verifies compressed as well as regular proofs; moreover, it transparently verifies both per-spec compressed proofs and the flawed format he uncovered (see comment below of 16-Oct-05).
(16-Oct-2005) Marnix Klooster noticed that for large proofs, the compressed proof format did not match the spec in the book. His algorithm to correct the problem has been put into the Metamath program (version 0.07.6). The program still verifies older proofs with the incorrect format, but the user will be nagged to update them with 'save proof *'. In set.mm, 285 out of 6376 proofs are affected. (The incorrect format did not affect proof correctness or verification, since the compression and decompression algorithms matched each other.)
(13-Sep-2005) Scott Fenton found an interesting axiom, ax46, which could be used to replace both ax-4 and ax-6.
(29-Jul-2005) Metamath was selected as site of the week by American Scientist Online.
(8-Jul-2005) Roy Longton has contributed 53 new theorems to the Quantum Logic Explorer. You can see them in the Theorem List starting at lem3.3.3lem1. He writes, "If you want, you can post an open challenge to see if anyone can find shorter proofs of the theorems I submitted."
(10-May-2005) A Usenet post I posted about the infinite prime proof; another one about indexed unions.
(3-May-2005) The theorem divexpt is the 5,000th theorem added to the Metamath Proof Explorer database.
(12-Apr-2005) Raph Levien solved the open problem in item 16 on the Workshop Miscellany page and as a corollary proved that axiom ax-9 is independent from the other axioms of predicate calculus and equality. This is the first such independence proof so far; a goal is to prove all of them independent (or to derive any redundant ones from the others).
(8-Mar-2005) I added a paragraph above our complex number axioms table, summarizing the construction and indicating where Dedekind cuts are defined. Thanks to Andrew Buhr for comments on this.
(16-Feb-2005) The Metamath Music Page is mentioned as a reference or resource for a university course called Math, Mind, and Music. .
(28-Jan-2005) Steven Cullinane parodied the Metamath Music Page in his blog.
(18-Jan-2005) Waldek Hebisch upgraded the Metamath program to run on the AMD64 64-bit processor.
(17-Jan-2005) A symbol list summary was added to the beginning of the Hilbert Space Explorer Home Page. Thanks to Mladen Pavicic for suggesting this.
(6-Jan-2005) Someone assembled an amazon.com list of some of the books in the Metamath Proof Explorer Bibliography.
(4-Jan-2005) The definition of ordinal exponentiation was decided on after this Usenet discussion.
(19-Dec-2004) A bit of trivia: my Erdös number is 2, as you can see from this list.
(20-Oct-2004) I started this Usenet discussion about the "reals are uncountable" proof (127 comments; last one on Nov. 12).
(12-Oct-2004) gch-kn shows the equivalence of the Generalized Continuum Hypothesis and Prof. Nambiar's Axiom of Combinatorial Sets. This proof answers his Open Problem 2 (PDF file).
(5-Aug-2004) I gave a talk on "Hilbert Lattice Equations" at the Argonne workshop.
(25-Jul-2004) The theorem nthruz is the 4,000th theorem added to the Metamath Proof Explorer database.
(27-May-2004) Josiah Burroughs contributed the proofs u1lemn1b, u1lem3var1, oi3oa3lem1, and oi3oa3 to the Quantum Logic Explorer database ql.mm.
(23-May-2004) Some minor typos found by Josh Purinton were corrected in the Metamath book. In addition, Josh simplified the definition of the closure of a pre-statement of a formal system in Appendix C.
(5-May-2004) Gregory Bush has found shorter proofs for 67 of the 193 propositional calculus theorems listed in Principia Mathematica, thus establishing 67 new records. (This was challenge #4 on the open problems page.)
Copyright terms: Public domain | W3C HTML validation [external] |