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Theorem List for Metamath Proof Explorer - 12601-12700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxleneg 12601 Extended real version of leneg 11132. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴))
 
Theoremxlt0neg1 12602 Extended real version of lt0neg1 11135. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴))
 
Theoremxlt0neg2 12603 Extended real version of lt0neg2 11136. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0))
 
Theoremxle0neg1 12604 Extended real version of le0neg1 11137. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴))
 
Theoremxle0neg2 12605 Extended real version of le0neg2 11138. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (0 ≤ 𝐴 ↔ -𝑒𝐴 ≤ 0))
 
Theoremxaddval 12606 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))))))
 
Theoremxaddf 12607 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
+𝑒 :(ℝ* × ℝ*)⟶ℝ*
 
Theoremxmulval 12608 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 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
 
Theoremxaddpnf1 12609 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)
 
Theoremxaddpnf2 12610 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞)
 
Theoremxaddmnf1 12611 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)
 
Theoremxaddmnf2 12612 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞)
 
Theorempnfaddmnf 12613 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
 
Theoremmnfaddpnf 12614 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
 
Theoremrexadd 12615 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))
 
Theoremrexsub 12616 Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 -𝑒𝐵) = (𝐴𝐵))
 
Theoremrexaddd 12617 The extended real addition operation when both arguments are real. Deduction version of rexadd 12615. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))
 
Theoremxnn0xaddcl 12618 The extended nonnegative integers are closed under extended addition. (Contributed by AV, 10-Dec-2020.)
((𝐴 ∈ ℕ0*𝐵 ∈ ℕ0*) → (𝐴 +𝑒 𝐵) ∈ ℕ0*)
 
Theoremxaddnemnf 12619 Closure of extended real addition in the subset * / {-∞}. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ*𝐴 ≠ -∞) ∧ (𝐵 ∈ ℝ*𝐵 ≠ -∞)) → (𝐴 +𝑒 𝐵) ≠ -∞)
 
Theoremxaddnepnf 12620 Closure of extended real addition in the subset * / {+∞}. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ*𝐴 ≠ +∞) ∧ (𝐵 ∈ ℝ*𝐵 ≠ +∞)) → (𝐴 +𝑒 𝐵) ≠ +∞)
 
Theoremxnegid 12621 Extended real version of negid 10922. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0)
 
Theoremxaddcl 12622 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*)
 
Theoremxaddcom 12623 The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = (𝐵 +𝑒 𝐴))
 
Theoremxaddid1 12624 Extended real version of addid1 10809. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴)
 
Theoremxaddid2 12625 Extended real version of addid2 10812. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴)
 
Theoremxaddid1d 12626 0 is a right identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → (𝐴 +𝑒 0) = 𝐴)
 
Theoremxnn0lem1lt 12627 Extended nonnegative integer ordering relation. (Contributed by Thierry Arnoux, 30-Jul-2023.)
((𝑀 ∈ ℕ0𝑁 ∈ ℕ0*) → (𝑀𝑁 ↔ (𝑀 − 1) < 𝑁))
 
Theoremxnn0lenn0nn0 12628 An extended nonnegative integer which is less than or equal to a nonnegative integer is a nonnegative integer. (Contributed by AV, 24-Nov-2021.)
((𝑀 ∈ ℕ0*𝑁 ∈ ℕ0𝑀𝑁) → 𝑀 ∈ ℕ0)
 
Theoremxnn0le2is012 12629 An extended nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 24-Nov-2021.)
((𝑁 ∈ ℕ0*𝑁 ≤ 2) → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))
 
Theoremxnn0xadd0 12630 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)))
 
Theoremxnegdi 12631 Extended real version of negdi 10932. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 𝐵) = (-𝑒𝐴 +𝑒 -𝑒𝐵))
 
Theoremxaddass 12632 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 12633, 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)))
 
Theoremxaddass2 12633 Associativity of extended real addition. See xaddass 12632 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)))
 
Theoremxpncan 12634 Extended real version of pncan 10881. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → ((𝐴 +𝑒 𝐵) +𝑒 -𝑒𝐵) = 𝐴)
 
Theoremxnpcan 12635 Extended real version of npcan 10884. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)
 
Theoremxleadd1a 12636 Extended real version of leadd1 11097; 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*) ∧ 𝐴𝐵) → (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶))
 
Theoremxleadd2a 12637 Commuted form of xleadd1a 12636. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ 𝐴𝐵) → (𝐶 +𝑒 𝐴) ≤ (𝐶 +𝑒 𝐵))
 
Theoremxleadd1 12638 Weakened version of xleadd1a 12636 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ) → (𝐴𝐵 ↔ (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶)))
 
Theoremxltadd1 12639 Extended real version of ltadd1 11096. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 +𝑒 𝐶) < (𝐵 +𝑒 𝐶)))
 
Theoremxltadd2 12640 Extended real version of ltadd2 10733. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 +𝑒 𝐴) < (𝐶 +𝑒 𝐵)))
 
Theoremxaddge0 12641 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))
 
Theoremxle2add 12642 Extended real version of le2add 11111. (Contributed by Mario Carneiro, 23-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴𝐶𝐵𝐷) → (𝐴 +𝑒 𝐵) ≤ (𝐶 +𝑒 𝐷)))
 
Theoremxlt2add 12643 Extended real version of lt2add 11114. Note that ltleadd 11112, 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 < 𝐶𝐵 < 𝐷) → (𝐴 +𝑒 𝐵) < (𝐶 +𝑒 𝐷)))
 
Theoremxsubge0 12644 Extended real version of subge0 11142. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵𝐴))
 
Theoremxposdif 12645 Extended real version of posdif 11122. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +𝑒 -𝑒𝐴)))
 
Theoremxlesubadd 12646 Under certain conditions, the conclusion of lesubadd 11101 is true even in the extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (0 ≤ 𝐴𝐵 ≠ -∞ ∧ 0 ≤ 𝐶)) → ((𝐴 +𝑒 -𝑒𝐵) ≤ 𝐶𝐴 ≤ (𝐶 +𝑒 𝐵)))
 
Theoremxmullem 12647 Lemma for rexmul 12654. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) ∧ ¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → 𝐴 ∈ ℝ)
 
Theoremxmullem2 12648 Lemma for xmulneg1 12652. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))
 
Theoremxmulcom 12649 Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) = (𝐵 ·e 𝐴))
 
Theoremxmul01 12650 Extended real version of mul01 10808. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0)
 
Theoremxmul02 12651 Extended real version of mul02 10807. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 ·e 𝐴) = 0)
 
Theoremxmulneg1 12652 Extended real version of mulneg1 11065. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = -𝑒(𝐴 ·e 𝐵))
 
Theoremxmulneg2 12653 Extended real version of mulneg2 11066. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e -𝑒𝐵) = -𝑒(𝐴 ·e 𝐵))
 
Theoremrexmul 12654 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))
 
Theoremxmulf 12655 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
·e :(ℝ* × ℝ*)⟶ℝ*
 
Theoremxmulcl 12656 Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) ∈ ℝ*)
 
Theoremxmulpnf1 12657 Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e +∞) = +∞)
 
Theoremxmulpnf2 12658 Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (+∞ ·e 𝐴) = +∞)
 
Theoremxmulmnf1 12659 Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e -∞) = -∞)
 
Theoremxmulmnf2 12660 Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (-∞ ·e 𝐴) = -∞)
 
Theoremxmulpnf1n 12661 Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 < 0) → (𝐴 ·e +∞) = -∞)
 
Theoremxmulid1 12662 Extended real version of mulid1 10628. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 ·e 1) = 𝐴)
 
Theoremxmulid2 12663 Extended real version of mulid2 10629. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (1 ·e 𝐴) = 𝐴)
 
Theoremxmulm1 12664 Extended real version of mulm1 11070. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (-1 ·e 𝐴) = -𝑒𝐴)
 
Theoremxmulasslem2 12665 Lemma for xmulass 12670. (Contributed by Mario Carneiro, 20-Aug-2015.)
((0 < 𝐴𝐴 = -∞) → 𝜑)
 
Theoremxmulgt0 12666 Extended real version of mulgt0 10707. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 < 𝐵)) → 0 < (𝐴 ·e 𝐵))
 
Theoremxmulge0 12667 Extended real version of mulge0 11147. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 ·e 𝐵))
 
Theoremxmulasslem 12668* Lemma for xmulass 12670. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝑥 = 𝐷 → (𝜓𝑋 = 𝑌))    &   (𝑥 = -𝑒𝐷 → (𝜓𝐸 = 𝐹))    &   (𝜑𝑋 ∈ ℝ*)    &   (𝜑𝑌 ∈ ℝ*)    &   (𝜑𝐷 ∈ ℝ*)    &   ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝜓)    &   (𝜑 → (𝑥 = 0 → 𝜓))    &   (𝜑𝐸 = -𝑒𝑋)    &   (𝜑𝐹 = -𝑒𝑌)       (𝜑𝑋 = 𝑌)
 
Theoremxmulasslem3 12669 Lemma for xmulass 12670. (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)))
 
Theoremxmulass 12670 Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 12632 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)))
 
Theoremxlemul1a 12671 Extended real version of lemul1a 11483. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶)) ∧ 𝐴𝐵) → (𝐴 ·e 𝐶) ≤ (𝐵 ·e 𝐶))
 
Theoremxlemul2a 12672 Extended real version of lemul2a 11484. (Contributed by Mario Carneiro, 8-Sep-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶)) ∧ 𝐴𝐵) → (𝐶 ·e 𝐴) ≤ (𝐶 ·e 𝐵))
 
Theoremxlemul1 12673 Extended real version of lemul1 11481. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ+) → (𝐴𝐵 ↔ (𝐴 ·e 𝐶) ≤ (𝐵 ·e 𝐶)))
 
Theoremxlemul2 12674 Extended real version of lemul2 11482. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ+) → (𝐴𝐵 ↔ (𝐶 ·e 𝐴) ≤ (𝐶 ·e 𝐵)))
 
Theoremxltmul1 12675 Extended real version of ltmul1 11479. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐴 ·e 𝐶) < (𝐵 ·e 𝐶)))
 
Theoremxltmul2 12676 Extended real version of ltmul2 11480. (Contributed by Mario Carneiro, 8-Sep-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐶 ·e 𝐴) < (𝐶 ·e 𝐵)))
 
Theoremxadddilem 12677 Lemma for xadddi 12678. (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)))
 
Theoremxadddi 12678 Distributive property for extended real addition and multiplication. Like xaddass 12632, 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)))
 
Theoremxadddir 12679 Commuted version of xadddi 12678. (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)))
 
Theoremxadddi2 12680 The assumption that the multiplier be real in xadddi 12678 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)))
 
Theoremxadddi2r 12681 Commuted version of xadddi2 12680. (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)))
 
Theoremx2times 12682 Extended real version of 2times 11762. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (2 ·e 𝐴) = (𝐴 +𝑒 𝐴))
 
Theoremxnegcld 12683 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → -𝑒𝐴 ∈ ℝ*)
 
Theoremxaddcld 12684 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐴 +𝑒 𝐵) ∈ ℝ*)
 
Theoremxmulcld 12685 Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐴 ·e 𝐵) ∈ ℝ*)
 
Theoremxadd4d 12686 Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 10857. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
(𝜑 → (𝐴 ∈ ℝ*𝐴 ≠ -∞))    &   (𝜑 → (𝐵 ∈ ℝ*𝐵 ≠ -∞))    &   (𝜑 → (𝐶 ∈ ℝ*𝐶 ≠ -∞))    &   (𝜑 → (𝐷 ∈ ℝ*𝐷 ≠ -∞))       (𝜑 → ((𝐴 +𝑒 𝐵) +𝑒 (𝐶 +𝑒 𝐷)) = ((𝐴 +𝑒 𝐶) +𝑒 (𝐵 +𝑒 𝐷)))
 
Theoremxnn0add4d 12687 Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 12686. (Contributed by AV, 12-Dec-2020.)
(𝜑𝐴 ∈ ℕ0*)    &   (𝜑𝐵 ∈ ℕ0*)    &   (𝜑𝐶 ∈ ℕ0*)    &   (𝜑𝐷 ∈ ℕ0*)       (𝜑 → ((𝐴 +𝑒 𝐵) +𝑒 (𝐶 +𝑒 𝐷)) = ((𝐴 +𝑒 𝐶) +𝑒 (𝐵 +𝑒 𝐷)))
 
5.5.3  Supremum and infimum on the extended reals
 
Theoremxrsupexmnf 12688* Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.)
(∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)))
 
Theoremxrinfmexpnf 12689* Adding plus infinity to a set does not affect the existence of its infimum. (Contributed by NM, 19-Jan-2006.)
(∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦)))
 
Theoremxrsupsslem 12690* Lemma for xrsupss 12692. (Contributed by NM, 25-Oct-2005.)
((𝐴 ⊆ ℝ* ∧ (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴)) → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
 
Theoremxrinfmsslem 12691* Lemma for xrinfmss 12693. (Contributed by NM, 19-Jan-2006.)
((𝐴 ⊆ ℝ* ∧ (𝐴 ⊆ ℝ ∨ -∞ ∈ 𝐴)) → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
 
Theoremxrsupss 12692* Any subset of extended reals has a supremum. (Contributed by NM, 25-Oct-2005.)
(𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
 
Theoremxrinfmss 12693* Any subset of extended reals has an infimum. (Contributed by NM, 25-Oct-2005.)
(𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
 
Theoremxrinfmss2 12694* Any subset of extended reals has an infimum. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
 
Theoremxrub 12695* By quantifying only over reals, we can specify any extended real upper bound for any set of extended reals. (Contributed by NM, 9-Apr-2006.)
((𝐴 ⊆ ℝ*𝐵 ∈ ℝ*) → (∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦𝐴 𝑥 < 𝑦) ↔ ∀𝑥 ∈ ℝ* (𝑥 < 𝐵 → ∃𝑦𝐴 𝑥 < 𝑦)))
 
Theoremsupxr 12696* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.) (Revised by Mario Carneiro, 21-Apr-2015.)
(((𝐴 ⊆ ℝ*𝐵 ∈ ℝ*) ∧ (∀𝑥𝐴 ¬ 𝐵 < 𝑥 ∧ ∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦𝐴 𝑥 < 𝑦))) → sup(𝐴, ℝ*, < ) = 𝐵)
 
Theoremsupxr2 12697* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.)
(((𝐴 ⊆ ℝ*𝐵 ∈ ℝ*) ∧ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦𝐴 𝑥 < 𝑦))) → sup(𝐴, ℝ*, < ) = 𝐵)
 
Theoremsupxrcl 12698 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 24-Oct-2005.)
(𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*)
 
Theoremsupxrun 12699 The supremum of the union of two sets of extended reals equals the largest of their suprema. (Contributed by NM, 19-Jan-2006.)
((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ* ∧ sup(𝐴, ℝ*, < ) ≤ sup(𝐵, ℝ*, < )) → sup((𝐴𝐵), ℝ*, < ) = sup(𝐵, ℝ*, < ))
 
Theoremsupxrmnf 12700 Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.)
(𝐴 ⊆ ℝ* → sup((𝐴 ∪ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))
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