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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | ccp 30801 | Metric completion of _Qp. |
class Cp | ||
Definition | df-zrng 30802 | Define the subring of integral elements in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ ZRing = (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟))) | ||
Definition | df-gf 30803* | Define the Galois finite field of order 𝑝↑𝑛. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ GF = (𝑝 ∈ ℙ, 𝑛 ∈ ℕ ↦ ⦋(ℤ/nℤ‘𝑝) / 𝑟⦌(1st ‘(𝑟 splitFld {⦋(Poly1‘𝑟) / 𝑠⦌⦋(var1‘𝑟) / 𝑥⦌(((𝑝↑𝑛)(.g‘(mulGrp‘𝑠))𝑥)(-g‘𝑠)𝑥)}))) | ||
Definition | df-gfoo 30804* | Define the Galois field of order 𝑝↑+∞, as a direct limit of the Galois finite fields. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ GF∞ = (𝑝 ∈ ℙ ↦ ⦋(ℤ/nℤ‘𝑝) / 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {⦋(Poly1‘𝑟) / 𝑠⦌⦋(var1‘𝑟) / 𝑥⦌(((𝑝↑𝑛)(.g‘(mulGrp‘𝑠))𝑥)(-g‘𝑠)𝑥)}))) | ||
Definition | df-eqp 30805* | Define an equivalence relation on ℤ-indexed sequences of integers such that two sequences are equivalent iff the difference is equivalent to zero, and a sequence is equivalent to zero iff the sum Σ𝑘 ≤ 𝑛𝑓(𝑘)(𝑝↑𝑘) is a multiple of 𝑝↑(𝑛 + 1) for every 𝑛. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ ~Qp = (𝑝 ∈ ℙ ↦ {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ (ℤ ↑𝑚 ℤ) ∧ ∀𝑛 ∈ ℤ Σ𝑘 ∈ (ℤ≥‘-𝑛)(((𝑓‘-𝑘) − (𝑔‘-𝑘)) / (𝑝↑(𝑘 + (𝑛 + 1)))) ∈ ℤ)}) | ||
Definition | df-rqp 30806* | There is a unique element of (ℤ ↑𝑚 (0...(𝑝 − 1))) ~Qp -equivalent to any element of (ℤ ↑𝑚 ℤ), if the sequences are zero for sufficiently large negative values; this function selects that element. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ /Qp = (𝑝 ∈ ℙ ↦ (~Qp ∩ ⦋{𝑓 ∈ (ℤ ↑𝑚 ℤ) ∣ ∃𝑥 ∈ ran ℤ≥(◡𝑓 “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑦⦌(𝑦 × (𝑦 ∩ (ℤ ↑𝑚 (0...(𝑝 − 1))))))) | ||
Definition | 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, < )))))) | ||
Definition | df-qpOLD 30808* | 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.) Obsolete version of df-qp 30807 as of 10-Oct-2021. (New usage is discouraged.) |
⊢ QpOLD = (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↑-sup((◡𝑓 “ (ℤ ∖ {0})), ℝ, ◡ < )))))) | ||
Definition | df-zp 30809 | Define the 𝑝-adic integers, as a subset of the 𝑝-adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ Zp = (ZRing ∘ Qp) | ||
Definition | df-qpa 30810* | Define the completion of the 𝑝-adic rationals. Here we simply define it as the splitting field of a dense sequence of polynomials (using as the 𝑛-th set the collection of polynomials with degree less than 𝑛 and with coefficients < (𝑝↑𝑛)). Krasner's lemma will then show that all monic polynomials have splitting fields isomorphic to a sufficiently close Eisenstein polynomial from the list, and unramified extensions are generated by the polynomial 𝑥↑(𝑝↑𝑛) − 𝑥, which is in the list. Thus, every finite extension of Qp is a subfield of this field extension, so it is algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ _Qp = (𝑝 ∈ ℙ ↦ ⦋(Qp‘𝑝) / 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {𝑓 ∈ (Poly1‘𝑟) ∣ ((𝑟 deg1 𝑓) ≤ 𝑛 ∧ ∀𝑑 ∈ ran (coe1‘𝑓)(◡𝑑 “ (ℤ ∖ {0})) ⊆ (0...𝑛))}))) | ||
Definition | df-cp 30811 | Define the metric completion of the algebraic completion of the 𝑝 -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ Cp = ( cplMetSp ∘ _Qp) | ||
I hope someone will enjoy solving (proving) the simple equations, inequalities, and calculations from this mathbox. I have proved these problems (theorems) using the Milpgame proof assistant. (It can be downloaded from http://us.metamath.org/other/milpgame/milpgame.html.) | ||
Theorem | problem1 30812 | Practice problem 1. Clues: 5p4e9 11044 3p2e5 11037 eqtri 2632 oveq1i 6559. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
⊢ ((3 + 2) + 4) = 9 | ||
Theorem | 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) | ||
Theorem | problem2OLD 30814 | 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.) Obsolete version of problem2 30813 as of 9-Sep-2021. (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((2 · 10) + 5) + ((1 · 10) + 4)) = ((3 · 10) + 9) | ||
Theorem | problem3 30815 | Practice problem 3. Clues: eqcomi 2619 eqtri 2632 subaddrii 10249 recni 9931 4re 10974 3re 10971 1re 9918 df-4 10958 addcomi 10106. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ ℂ & ⊢ (𝐴 + 3) = 4 ⇒ ⊢ 𝐴 = 1 | ||
Theorem | problem4 30816 | Practice problem 4. Clues: pm3.2i 470 eqcomi 2619 eqtri 2632 subaddrii 10249 recni 9931 7re 10980 6re 10978 ax-1cn 9873 df-7 10961 ax-mp 5 oveq1i 6559 3cn 10972 2cn 10968 df-3 10957 mulid2i 9922 subdiri 10359 mp3an 1416 mulcli 9924 subadd23 10172 oveq2i 6560 oveq12i 6561 3t2e6 11056 mulcomi 9925 subcli 10236 biimpri 217 subadd2i 10248. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ (𝐴 + 𝐵) = 3 & ⊢ ((3 · 𝐴) + (2 · 𝐵)) = 7 ⇒ ⊢ (𝐴 = 1 ∧ 𝐵 = 2) | ||
Theorem | problem5 30817 | Practice problem 5. Clues: 3brtr3i 4612 mpbi 219 breqtri 4608 ltaddsubi 10468 remulcli 9933 2re 10967 3re 10971 9re 10984 eqcomi 2619 mvlladdi 10178 3cn 6cn 10979 eqtr3i 2634 6p3e9 11047 addcomi 10106 ltdiv1ii 10832 6re 10978 nngt0i 10931 2nn 11062 divcan3i 10650 recni 9931 2cn 10968 2ne0 10990 mpbir 220 eqtri 2632 mulcomi 9925 3t2e6 11056 divmuli 10658. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ ℝ & ⊢ ((2 · 𝐴) + 3) < 9 ⇒ ⊢ 𝐴 < 3 | ||
Theorem | quad3 30818 | Variant of quadratic equation with discriminant expanded. (Contributed by Filip Cernatescu, 19-Oct-2019.) |
⊢ 𝑋 ∈ ℂ & ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠ 0 & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0 ⇒ ⊢ (𝑋 = ((-𝐵 + (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)) ∨ 𝑋 = ((-𝐵 − (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴))) | ||
Theorem | climuzcnv 30819* | Utility lemma to convert between 𝑚 ≤ 𝑘 and 𝑘 ∈ (ℤ≥‘𝑚) in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.) |
⊢ (𝑚 ∈ ℕ → ((𝑘 ∈ (ℤ≥‘𝑚) → 𝜑) ↔ (𝑘 ∈ ℕ → (𝑚 ≤ 𝑘 → 𝜑)))) | ||
Theorem | sinccvglem 30820* | ((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, 21-May-2014.) |
⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ∖ {0})) & ⊢ (𝜑 → 𝐹 ⇝ 0) & ⊢ 𝐺 = (𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥)) & ⊢ 𝐻 = (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘𝑘)) < 1) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) ⇝ 1) | ||
Theorem | sinccvg 30821* | ((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.) |
⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝐹 ⇝ 0) → ((𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥)) ∘ 𝐹) ⇝ 1) | ||
Theorem | circum 30822* | The circumference of a circle of radius 𝑅, defined as the limit as 𝑛 ⇝ +∞ of the perimeter of an inscribed n-sided isogons, is ((2 · π) · 𝑅). (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.) |
⊢ 𝐴 = ((2 · π) / 𝑛) & ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ ((2 · 𝑛) · (𝑅 · (sin‘(𝐴 / 2))))) & ⊢ 𝑅 ∈ ℝ ⇒ ⊢ 𝑃 ⇝ ((2 · π) · 𝑅) | ||
Theorem | elfzm12 30823 | Membership in a curtailed finite sequence of integers. (Contributed by Paul Chapman, 17-Nov-2012.) |
⊢ (𝑁 ∈ ℕ → (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ (1...𝑁))) | ||
Theorem | nn0seqcvg 30824* | A strictly-decreasing nonnegative integer sequence with initial term 𝑁 reaches zero by the 𝑁 th term. Inference version. (Contributed by Paul Chapman, 31-Mar-2011.) |
⊢ 𝐹:ℕ0⟶ℕ0 & ⊢ 𝑁 = (𝐹‘0) & ⊢ (𝑘 ∈ ℕ0 → ((𝐹‘(𝑘 + 1)) ≠ 0 → (𝐹‘(𝑘 + 1)) < (𝐹‘𝑘))) ⇒ ⊢ (𝐹‘𝑁) = 0 | ||
Theorem | lediv2aALT 30825 | Division of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) → (𝐴 ≤ 𝐵 → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴))) | ||
Theorem | abs2sqlei 30826 | The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2)) | ||
Theorem | abs2sqlti 30827 | The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2)) | ||
Theorem | abs2sqle 30828 | The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2))) | ||
Theorem | abs2sqlt 30829 | The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2))) | ||
Theorem | abs2difi 30830 | Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵)) | ||
Theorem | abs2difabsi 30831 | Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴 − 𝐵)) | ||
Theorem | axextprim 30832 | ax-ext 2590 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
⊢ ¬ ∀𝑥 ¬ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧)) | ||
Theorem | axrepprim 30833 | ax-rep 4699 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
⊢ ¬ ∀𝑥 ¬ (¬ ∀𝑦 ¬ ∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧 ∈ 𝑥 → ¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧 ∈ 𝑥))) | ||
Theorem | axunprim 30834 | ax-un 6847 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
⊢ ¬ ∀𝑥 ¬ ∀𝑦(¬ ∀𝑥(𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) | ||
Theorem | axpowprim 30835 | ax-pow 4769 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
⊢ (∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) → 𝑥 = 𝑦) | ||
Theorem | axregprim 30836 | ax-reg 8380 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
⊢ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) | ||
Theorem | axinfprim 30837 | ax-inf 8418 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 13-Oct-2010.) |
⊢ ¬ ∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ (𝑦 ∈ 𝑥 → ¬ ∀𝑦(𝑦 ∈ 𝑥 → ¬ ∀𝑧(𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥)))) | ||
Theorem | axacprim 30838 | ax-ac 9164 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 26-Oct-2010.) |
⊢ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥)))))) | ||
Theorem | untelirr 30839* | We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 30941). Using this concept, we can avoid a lot of the uses of the Axiom of Regularity. Here, we prove a series of properties of untanged classes. First, we prove that an untangled class is not a member of itself. (Contributed by Scott Fenton, 28-Feb-2011.) |
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴) | ||
Theorem | untuni 30840* | The union of a class is untangled iff all its members are untangled. (Contributed by Scott Fenton, 28-Feb-2011.) |
⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥) | ||
Theorem | untsucf 30841* | If a class is untangled, then so is its successor. (Contributed by Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.) |
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦) | ||
Theorem | unt0 30842 | The null set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ∀𝑥 ∈ ∅ ¬ 𝑥 ∈ 𝑥 | ||
Theorem | untint 30843* | If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.) |
⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦) | ||
Theorem | efrunt 30844* | If 𝐴 is well-founded by E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.) |
⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥) | ||
Theorem | untangtr 30845* | A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.) |
⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦)) | ||
Theorem | 3orel1 30846 | Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.) |
⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓 ∨ 𝜒) → (𝜓 ∨ 𝜒))) | ||
Theorem | 3orel2 30847 | Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓 ∨ 𝜒) → (𝜑 ∨ 𝜒))) | ||
Theorem | 3orel3 30848 | Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.) |
⊢ (¬ 𝜒 → ((𝜑 ∨ 𝜓 ∨ 𝜒) → (𝜑 ∨ 𝜓))) | ||
Theorem | 3pm3.2ni 30849 | Triple negated disjunction introduction. (Contributed by Scott Fenton, 20-Apr-2011.) |
⊢ ¬ 𝜑 & ⊢ ¬ 𝜓 & ⊢ ¬ 𝜒 ⇒ ⊢ ¬ (𝜑 ∨ 𝜓 ∨ 𝜒) | ||
Theorem | 3jaodd 30850 | Double deduction form of 3jaoi 1383. (Contributed by Scott Fenton, 20-Apr-2011.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) & ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂))) & ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜃 ∨ 𝜏) → 𝜂))) | ||
Theorem | 3orit 30851 | Closed form of 3ori 1380. (Contributed by Scott Fenton, 20-Apr-2011.) |
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)) | ||
Theorem | biimpexp 30852 | A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.) |
⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜒))) | ||
Theorem | 3orel13 30853 | Elimination of two disjuncts in a triple disjunction. (Contributed by Scott Fenton, 9-Jun-2011.) |
⊢ ((¬ 𝜑 ∧ ¬ 𝜒) → ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜓)) | ||
Theorem | nepss 30854 | Two classes are inequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.) |
⊢ (𝐴 ≠ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵)) | ||
Theorem | 3ccased 30855 | Triple disjunction form of ccased 985. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (𝜑 → ((𝜒 ∧ 𝜂) → 𝜓)) & ⊢ (𝜑 → ((𝜒 ∧ 𝜁) → 𝜓)) & ⊢ (𝜑 → ((𝜒 ∧ 𝜎) → 𝜓)) & ⊢ (𝜑 → ((𝜃 ∧ 𝜂) → 𝜓)) & ⊢ (𝜑 → ((𝜃 ∧ 𝜁) → 𝜓)) & ⊢ (𝜑 → ((𝜃 ∧ 𝜎) → 𝜓)) & ⊢ (𝜑 → ((𝜏 ∧ 𝜂) → 𝜓)) & ⊢ (𝜑 → ((𝜏 ∧ 𝜁) → 𝜓)) & ⊢ (𝜑 → ((𝜏 ∧ 𝜎) → 𝜓)) ⇒ ⊢ (𝜑 → (((𝜒 ∨ 𝜃 ∨ 𝜏) ∧ (𝜂 ∨ 𝜁 ∨ 𝜎)) → 𝜓)) | ||
Theorem | dfso3 30856* | Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.) |
⊢ (𝑅 Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | ||
Theorem | brtpid1 30857 | A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.) |
⊢ 𝐴{〈𝐴, 𝐵〉, 𝐶, 𝐷}𝐵 | ||
Theorem | brtpid2 30858 | A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.) |
⊢ 𝐴{𝐶, 〈𝐴, 𝐵〉, 𝐷}𝐵 | ||
Theorem | brtpid3 30859 | A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.) |
⊢ 𝐴{𝐶, 𝐷, 〈𝐴, 𝐵〉}𝐵 | ||
Theorem | ceqsrexv2 30860* | Alternate elimitation of a restricted existential quantifier, using implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) | ||
Theorem | iota5f 30861* | A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴) | ||
Theorem | ceqsralv2 30862* | Alternate elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓)) | ||
Theorem | sqdivzi 30863 | Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) | ||
Theorem | subdivcomb1 30864 | Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((𝐶 · 𝐴) − 𝐵) / 𝐶) = (𝐴 − (𝐵 / 𝐶))) | ||
Theorem | subdivcomb2 30865 | Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 − (𝐶 · 𝐵)) / 𝐶) = ((𝐴 / 𝐶) − 𝐵)) | ||
Theorem | supfz 30866 | The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑁) | ||
Theorem | 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) | ||
Theorem | inffzOLD 30868 | The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) Obsolete version of inffz 30867 as of 10-Oct-2021. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, ◡ < ) = 𝑀) | ||
Theorem | fz0n 30869 | The sequence (0...(𝑁 − 1)) is empty iff 𝑁 is zero. (Contributed by Scott Fenton, 16-May-2014.) |
⊢ (𝑁 ∈ ℕ0 → ((0...(𝑁 − 1)) = ∅ ↔ 𝑁 = 0)) | ||
Theorem | shftvalg 30870 | Value of a sequence shifted by 𝐴. (Contributed by Scott Fenton, 16-Dec-2017.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴))) | ||
Theorem | divcnvlin 30871* | Limit of the ratio of two linear functions. (Contributed by Scott Fenton, 17-Dec-2017.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((𝑘 + 𝐴) / (𝑘 + 𝐵))) ⇒ ⊢ (𝜑 → 𝐹 ⇝ 1) | ||
Theorem | climlec3 30872* | Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | logi 30873 | Calculate the logarithm of i. (Contributed by Scott Fenton, 13-Apr-2020.) |
⊢ (log‘i) = (i · (π / 2)) | ||
Theorem | iexpire 30874 | i raised to itself is real. (Contributed by Scott Fenton, 13-Apr-2020.) |
⊢ (i↑𝑐i) ∈ ℝ | ||
Theorem | bcneg1 30875 | The binomial coefficent over negative one is zero. (Contributed by Scott Fenton, 29-May-2020.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0) | ||
Theorem | bcm1nt 30876 | The proportion of one bionmial coefficient to another with 𝑁 decreased by 1. (Contributed by Scott Fenton, 23-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (𝑁C𝐾) = (((𝑁 − 1)C𝐾) · (𝑁 / (𝑁 − 𝐾)))) | ||
Theorem | bcprod 30877* | A product identity for binomial coefficents. (Contributed by Scott Fenton, 23-Jun-2020.) |
⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ (1...(𝑁 − 1))((𝑁 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝑘↑((2 · 𝑘) − 𝑁))) | ||
Theorem | bccolsum 30878* | A column-sum rule for binomial coefficents. (Contributed by Scott Fenton, 24-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))) | ||
Theorem | iprodefisumlem 30879 | Lemma for iprodefisum 30880. (Contributed by Scott Fenton, 11-Feb-2018.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) ⇒ ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹))) | ||
Theorem | iprodefisum 30880* | Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (exp‘𝐵) = (exp‘Σ𝑘 ∈ 𝑍 𝐵)) | ||
Theorem | iprodgam 30881* | An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.) |
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) ⇒ ⊢ (𝜑 → (Γ‘𝐴) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴)) | ||
Theorem | faclimlem1 30882* | Lemma for faclim 30885. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) = (𝑥 ∈ ℕ ↦ ((𝑀 + 1) · ((𝑥 + 1) / (𝑥 + (𝑀 + 1)))))) | ||
Theorem | faclimlem2 30883* | Lemma for faclim 30885. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) ⇝ (𝑀 + 1)) | ||
Theorem | faclimlem3 30884 | Lemma for faclim 30885. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (((1 + (1 / 𝐵))↑(𝑀 + 1)) / (1 + ((𝑀 + 1) / 𝐵))) = ((((1 + (1 / 𝐵))↑𝑀) / (1 + (𝑀 / 𝐵))) · (((1 + (𝑀 / 𝐵)) · (1 + (1 / 𝐵))) / (1 + ((𝑀 + 1) / 𝐵))))) | ||
Theorem | faclim 30885* | An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛)))) ⇒ ⊢ (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴)) | ||
Theorem | iprodfac 30886* | An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘)))) | ||
Theorem | faclim2 30887* | Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((!‘𝑛) · ((𝑛 + 1)↑𝑀)) / (!‘(𝑛 + 𝑀)))) ⇒ ⊢ (𝑀 ∈ ℕ0 → 𝐹 ⇝ 1) | ||
Theorem | pdivsq 30888 | Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ) → (𝑃 ∥ 𝑀 ↔ 𝑃 ∥ (𝑀↑2))) | ||
Theorem | dvdspw 30889 | Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐾 ∥ 𝑀 → 𝐾 ∥ (𝑀↑𝑁))) | ||
Theorem | gcd32 30890 | Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = ((𝐴 gcd 𝐶) gcd 𝐵)) | ||
Theorem | gcdabsorb 30891 | Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐵) = (𝐴 gcd 𝐵)) | ||
Theorem | brtp 30892 | A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.) |
⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V ⇒ ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷) ∨ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹))) | ||
Theorem | dftr6 30893 | A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E ))) | ||
Theorem | coep 30894* | Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥) | ||
Theorem | coepr 30895* | Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵) | ||
Theorem | dffr5 30896 | A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.) |
⊢ (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅))) | ||
Theorem | dfso2 30897 | Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.) |
⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅)))) | ||
Theorem | dfpo2 30898 | Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.) |
⊢ (𝑅 Po 𝐴 ↔ ((𝑅 ∩ ( I ↾ 𝐴)) = ∅ ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅)) | ||
Theorem | br8 30899* | Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁)) & ⊢ (𝑔 = 𝐺 → (𝜁 ↔ 𝜎)) & ⊢ (ℎ = 𝐻 → (𝜎 ↔ 𝜌)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {〈𝑝, 𝑞〉 ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = 〈〈𝑎, 𝑏〉, 〈𝑐, 𝑑〉〉 ∧ 𝑞 = 〈〈𝑒, 𝑓〉, 〈𝑔, ℎ〉〉 ∧ 𝜑)} ⇒ ⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉𝑅〈〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉〉 ↔ 𝜌)) | ||
Theorem | br6 30900* | Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {〈𝑝, 𝑞〉 ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 (𝑝 = 〈𝑎, 〈𝑏, 𝑐〉〉 ∧ 𝑞 = 〈𝑑, 〈𝑒, 𝑓〉〉 ∧ 𝜑)} ⇒ ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄) ∧ (𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄)) → (〈𝐴, 〈𝐵, 𝐶〉〉𝑅〈𝐷, 〈𝐸, 𝐹〉〉 ↔ 𝜁)) |
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