HomeTheorem List (p. 120 of 424)
GIF version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List
| Home |
Metamath
Proof Explorer Theorem List (p. 120 of 424) | < Previous Next > |
| Bad symbols? Try the
GIF version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | Metamath Proof Explorer
(1-27159) |
Hilbert Space Explorer
(27160-28684) |
Users' Mathboxes
(28685-42360) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | max0sub 11901 | Decompose a real number into positive and negative parts. (Contributed by Mario Carneiro, 6-Aug-2014.) |
| ⊢ (𝐴 ∈ ℝ → (if(0 ≤ 𝐴, 𝐴, 0) − if(0 ≤ -𝐴, -𝐴, 0)) = 𝐴) | ||
| Theorem | ifle 11902 | An if statement transforms an implication into an inequality of terms. (Contributed by Mario Carneiro, 31-Aug-2014.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) → if(𝜑, 𝐴, 𝐵) ≤ if(𝜓, 𝐴, 𝐵)) | ||
| Theorem | z2ge 11903* | There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005.) |
| ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ∃𝑘 ∈ ℤ (𝑀 ≤ 𝑘 ∧ 𝑁 ≤ 𝑘)) | ||
| Theorem | qbtwnre 11904* | The rational numbers are dense in ℝ: any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.) (Proof shortened by Mario Carneiro, 13-Jun-2014.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) | ||
| Theorem | qbtwnxr 11905* | The rational numbers are dense in ℝ*: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) | ||
| Theorem | qsqueeze 11906* | If a nonnegative real is less than any positive rational, it is zero. (Contributed by NM, 6-Feb-2007.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℚ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 = 0) | ||
| Theorem | qextltlem 11907* | Lemma for qextlt 11908 and qextle . (Contributed by Mario Carneiro, 3-Oct-2014.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ∧ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵)))) | ||
| Theorem | qextlt 11908* | An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))) | ||
| Theorem | qextle 11909* | An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℚ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵))) | ||
| Theorem | xralrple 11910* | Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Mario Carneiro, 12-Jun-2014.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + 𝑥))) | ||
| Theorem | alrple 11911* | Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Mario Carneiro, 12-Jun-2014.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + 𝑥))) | ||
| Theorem | xnegeq 11912 | Equality of two extended numbers with -𝑒 in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵) | ||
| Theorem | xnegex 11913 | A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ -𝑒𝐴 ∈ V | ||
| Theorem | xnegpnf 11914 | Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) |
| ⊢ -𝑒+∞ = -∞ | ||
| Theorem | xnegmnf 11915 | Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.) |
| ⊢ -𝑒-∞ = +∞ | ||
| Theorem | rexneg 11916 | Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | ||
| Theorem | xneg0 11917 | The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ -𝑒0 = 0 | ||
| Theorem | xnegcl 11918 | Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) | ||
| Theorem | xnegneg 11919 | Extended real version of negneg 10210. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) | ||
| Theorem | xneg11 11920 | Extended real version of neg11 10211. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵 ↔ 𝐴 = 𝐵)) | ||
| Theorem | xltnegi 11921 | Forward direction of xltneg 11922. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴) | ||
| Theorem | xltneg 11922 | Extended real version of ltneg 10407. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴)) | ||
| Theorem | xleneg 11923 | Extended real version of leneg 10410. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴)) | ||
| Theorem | xlt0neg1 11924 | Extended real version of lt0neg1 10413. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴)) | ||
| Theorem | xlt0neg2 11925 | Extended real version of lt0neg2 10414. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0)) | ||
| Theorem | xle0neg1 11926 | Extended real version of le0neg1 10415. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴)) | ||
| Theorem | xle0neg2 11927 | Extended real version of le0neg2 10416. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (0 ≤ 𝐴 ↔ -𝑒𝐴 ≤ 0)) | ||
| Theorem | xaddval 11928 | Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))))) | ||
| Theorem | xaddf 11929 | The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.) |
| ⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ* | ||
| Theorem | xmulval 11930 | Value of the extended real multiplication operation. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))) | ||
| Theorem | xaddpnf1 11931 | Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞) | ||
| Theorem | xaddpnf2 11932 | Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞) | ||
| Theorem | xaddmnf1 11933 | Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞) | ||
| Theorem | xaddmnf2 11934 | Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞) | ||
| Theorem | pnfaddmnf 11935 | Addition of positive and negative infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (+∞ +𝑒 -∞) = 0 | ||
| Theorem | mnfaddpnf 11936 | Addition of negative and positive infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (-∞ +𝑒 +∞) = 0 | ||
| Theorem | rexadd 11937 | The extended real addition operation when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) | ||
| Theorem | rexsub 11938 | Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 -𝑒𝐵) = (𝐴 − 𝐵)) | ||
| Theorem | rexaddd 11939 | The extended real addition operation when both arguments are real. Deduction version of rexadd 11937. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) | ||
| Theorem | xnn0xaddcl 11940 | The extended nonnegative integers are closed under extended addition. (Contributed by AV, 10-Dec-2020.) |
| ⊢ ((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → (𝐴 +𝑒 𝐵) ∈ ℕ0*) | ||
| Theorem | xaddnemnf 11941 | Closure of extended real addition in the subset ℝ* / {-∞}. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞)) → (𝐴 +𝑒 𝐵) ≠ -∞) | ||
| Theorem | xaddnepnf 11942 | Closure of extended real addition in the subset ℝ* / {+∞}. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ +∞)) → (𝐴 +𝑒 𝐵) ≠ +∞) | ||
| Theorem | xnegid 11943 | Extended real version of negid 10207. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0) | ||
| Theorem | xaddcl 11944 | The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) ∈ ℝ*) | ||
| Theorem | xaddcom 11945 | The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = (𝐵 +𝑒 𝐴)) | ||
| Theorem | xaddid1 11946 | Extended real version of addid1 10095. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) | ||
| Theorem | xaddid2 11947 | Extended real version of addid2 10098. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴) | ||
| Theorem | xaddid1d 11948 | 0 is a right identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐴 +𝑒 0) = 𝐴) | ||
| Theorem | xnn0xadd0 11949 | The sum of two extended nonnegative integers is 0 iff each of the two extended nonnegative integers is 0. (Contributed by AV, 14-Dec-2020.) |
| ⊢ ((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) | ||
| Theorem | xnegdi 11950 | Extended real version of xnegdi 11950. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 𝐵) = (-𝑒𝐴 +𝑒 -𝑒𝐵)) | ||
| Theorem | xaddass 11951 | Associativity of extended real addition. The correct condition here is "it is not the case that both +∞ and -∞ appear as one of 𝐴, 𝐵, 𝐶, i.e. ¬ {+∞, -∞} ⊆ {𝐴, 𝐵, 𝐶}", but this condition is difficult to work with, so we break the theorem into two parts: this one, where -∞ is not present in 𝐴, 𝐵, 𝐶, and xaddass2 11952, where +∞ is not present. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞) ∧ (𝐶 ∈ ℝ* ∧ 𝐶 ≠ -∞)) → ((𝐴 +𝑒 𝐵) +𝑒 𝐶) = (𝐴 +𝑒 (𝐵 +𝑒 𝐶))) | ||
| Theorem | xaddass2 11952 | Associativity of extended real addition. See xaddass 11951 for notes on the hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ +∞) ∧ (𝐶 ∈ ℝ* ∧ 𝐶 ≠ +∞)) → ((𝐴 +𝑒 𝐵) +𝑒 𝐶) = (𝐴 +𝑒 (𝐵 +𝑒 𝐶))) | ||
| Theorem | xpncan 11953 | Extended real version of pncan 10166. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 𝐵) +𝑒 -𝑒𝐵) = 𝐴) | ||
| Theorem | xnpcan 11954 | Extended real version of npcan 10169. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴) | ||
| Theorem | xleadd1a 11955 | Extended real version of leadd1 10375; note that the converse implication is not true, unlike the real version (for example 0 < 1 but (1 +𝑒 +∞) ≤ (0 +𝑒 +∞)). (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶)) | ||
| Theorem | xleadd2a 11956 | Commuted form of xleadd1a 11955. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐶 +𝑒 𝐴) ≤ (𝐶 +𝑒 𝐵)) | ||
| Theorem | xleadd1 11957 | Weakened version of xleadd1a 11955 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶))) | ||
| Theorem | xltadd1 11958 | Extended real version of ltadd1 10374. (Contributed by Mario Carneiro, 23-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 +𝑒 𝐶) < (𝐵 +𝑒 𝐶))) | ||
| Theorem | xltadd2 11959 | Extended real version of ltadd2 10020. (Contributed by Mario Carneiro, 23-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 +𝑒 𝐴) < (𝐶 +𝑒 𝐵))) | ||
| Theorem | xaddge0 11960 | The sum of nonnegative extended reals is nonnegative. (Contributed by Mario Carneiro, 21-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 +𝑒 𝐵)) | ||
| Theorem | xle2add 11961 | Extended real version of le2add 10389. (Contributed by Mario Carneiro, 23-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 +𝑒 𝐵) ≤ (𝐶 +𝑒 𝐷))) | ||
| Theorem | xlt2add 11962 | Extended real version of lt2add 10392. Note that ltleadd 10390, which has weaker assumptions, is not true for the extended reals (since 0 + +∞ < 1 + +∞ fails). (Contributed by Mario Carneiro, 23-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴 < 𝐶 ∧ 𝐵 < 𝐷) → (𝐴 +𝑒 𝐵) < (𝐶 +𝑒 𝐷))) | ||
| Theorem | xsubge0 11963 | Extended real version of subge0 10420. (Contributed by Mario Carneiro, 24-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴)) | ||
| Theorem | xposdif 11964 | Extended real version of posdif 10400. (Contributed by Mario Carneiro, 24-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +𝑒 -𝑒𝐴))) | ||
| Theorem | xlesubadd 11965 | Under certain conditions, the conclusion of lesubadd 10379 is true even in the extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (0 ≤ 𝐴 ∧ 𝐵 ≠ -∞ ∧ 0 ≤ 𝐶)) → ((𝐴 +𝑒 -𝑒𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 +𝑒 𝐵))) | ||
| Theorem | xmullem 11966 | Lemma for rexmul 11973. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) ∧ ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → 𝐴 ∈ ℝ) | ||
| Theorem | xmullem2 11967 | Lemma for xmulneg1 11971. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))) | ||
| Theorem | xmulcom 11968 | Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) = (𝐵 ·e 𝐴)) | ||
| Theorem | xmul01 11969 | Extended real version of mul01 10094. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0) | ||
| Theorem | xmul02 11970 | Extended real version of mul02 10093. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (0 ·e 𝐴) = 0) | ||
| Theorem | xmulneg1 11971 | Extended real version of mulneg1 10345. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = -𝑒(𝐴 ·e 𝐵)) | ||
| Theorem | xmulneg2 11972 | Extended real version of mulneg2 10346. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e -𝑒𝐵) = -𝑒(𝐴 ·e 𝐵)) | ||
| Theorem | rexmul 11973 | The extended real multiplication when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ·e 𝐵) = (𝐴 · 𝐵)) | ||
| Theorem | xmulf 11974 | The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.) |
| ⊢ ·e :(ℝ* × ℝ*)⟶ℝ* | ||
| Theorem | xmulcl 11975 | Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) ∈ ℝ*) | ||
| Theorem | xmulpnf1 11976 | Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e +∞) = +∞) | ||
| Theorem | xmulpnf2 11977 | Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (+∞ ·e 𝐴) = +∞) | ||
| Theorem | xmulmnf1 11978 | Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e -∞) = -∞) | ||
| Theorem | xmulmnf2 11979 | Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (-∞ ·e 𝐴) = -∞) | ||
| Theorem | xmulpnf1n 11980 | Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (𝐴 ·e +∞) = -∞) | ||
| Theorem | xmulid1 11981 | Extended real version of mulid1 9916. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 1) = 𝐴) | ||
| Theorem | xmulid2 11982 | Extended real version of mulid2 9917. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (1 ·e 𝐴) = 𝐴) | ||
| Theorem | xmulm1 11983 | Extended real version of mulm1 10350. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (-1 ·e 𝐴) = -𝑒𝐴) | ||
| Theorem | xmulasslem2 11984 | Lemma for xmulass 11989. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((0 < 𝐴 ∧ 𝐴 = -∞) → 𝜑) | ||
| Theorem | xmulgt0 11985 | Extended real version of mulgt0 9994. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 < 𝐵)) → 0 < (𝐴 ·e 𝐵)) | ||
| Theorem | xmulge0 11986 | Extended real version of mulge0 10425. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 ·e 𝐵)) | ||
| Theorem | xmulasslem 11987* | Lemma for xmulass 11989. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝑥 = 𝐷 → (𝜓 ↔ 𝑋 = 𝑌)) & ⊢ (𝑥 = -𝑒𝐷 → (𝜓 ↔ 𝐸 = 𝐹)) & ⊢ (𝜑 → 𝑋 ∈ ℝ*) & ⊢ (𝜑 → 𝑌 ∈ ℝ*) & ⊢ (𝜑 → 𝐷 ∈ ℝ*) & ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝜓) & ⊢ (𝜑 → (𝑥 = 0 → 𝜓)) & ⊢ (𝜑 → 𝐸 = -𝑒𝑋) & ⊢ (𝜑 → 𝐹 = -𝑒𝑌) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
| Theorem | xmulasslem3 11988 | Lemma for xmulass 11989. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ* ∧ 0 < 𝐶)) → ((𝐴 ·e 𝐵) ·e 𝐶) = (𝐴 ·e (𝐵 ·e 𝐶))) | ||
| Theorem | xmulass 11989 | Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 11951 which has to avoid the "undefined" combinations +∞ +𝑒 -∞ and -∞ +𝑒 +∞. The equivalent "undefined" expression here would be 0 ·e +∞, but since this is defined to equal 0 any zeroes in the expression make the whole thing evaluate to zero (on both sides), thus establishing the identity in this case. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ·e 𝐵) ·e 𝐶) = (𝐴 ·e (𝐵 ·e 𝐶))) | ||
| Theorem | xlemul1a 11990 | Extended real version of lemul1a 10756. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐴 ·e 𝐶) ≤ (𝐵 ·e 𝐶)) | ||
| Theorem | xlemul2a 11991 | Extended real version of lemul2a 10757. (Contributed by Mario Carneiro, 8-Sep-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐶 ·e 𝐴) ≤ (𝐶 ·e 𝐵)) | ||
| Theorem | xlemul1 11992 | Extended real version of lemul1 10754. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+) → (𝐴 ≤ 𝐵 ↔ (𝐴 ·e 𝐶) ≤ (𝐵 ·e 𝐶))) | ||
| Theorem | xlemul2 11993 | Extended real version of lemul2 10755. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+) → (𝐴 ≤ 𝐵 ↔ (𝐶 ·e 𝐴) ≤ (𝐶 ·e 𝐵))) | ||
| Theorem | xltmul1 11994 | Extended real version of ltmul1 10752. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐴 ·e 𝐶) < (𝐵 ·e 𝐶))) | ||
| Theorem | xltmul2 11995 | Extended real version of ltmul2 10753. (Contributed by Mario Carneiro, 8-Sep-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐶 ·e 𝐴) < (𝐶 ·e 𝐵))) | ||
| Theorem | xadddilem 11996 | Lemma for xadddi 11997. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 0 < 𝐴) → (𝐴 ·e (𝐵 +𝑒 𝐶)) = ((𝐴 ·e 𝐵) +𝑒 (𝐴 ·e 𝐶))) | ||
| Theorem | xadddi 11997 | Distributive property for extended real addition and multiplication. Like xaddass 11951, this has an unusual domain of correctness due to counterexamples like (+∞ · (2 − 1)) = -∞ ≠ ((+∞ · 2) − (+∞ · 1)) = (+∞ − +∞) = 0. In this theorem we show that if the multiplier is real then everything works as expected. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ·e (𝐵 +𝑒 𝐶)) = ((𝐴 ·e 𝐵) +𝑒 (𝐴 ·e 𝐶))) | ||
| Theorem | xadddir 11998 | Commuted version of xadddi 11997. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → ((𝐴 +𝑒 𝐵) ·e 𝐶) = ((𝐴 ·e 𝐶) +𝑒 (𝐵 ·e 𝐶))) | ||
| Theorem | xadddi2 11999 | The assumption that the multiplier be real in xadddi 11997 can be relaxed if the addends have the same sign. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶)) → (𝐴 ·e (𝐵 +𝑒 𝐶)) = ((𝐴 ·e 𝐵) +𝑒 (𝐴 ·e 𝐶))) | ||
| Theorem | xadddi2r 12000 | Commuted version of xadddi2 11999. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ*) → ((𝐴 +𝑒 𝐵) ·e 𝐶) = ((𝐴 ·e 𝐶) +𝑒 (𝐵 ·e 𝐶))) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |