P. 131 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13001-13100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xaddge0 13001	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 13002	Extended real version of le2add 11466. (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 13003	Extended real version of lt2add 11469. Note that ltleadd 11467, 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 13004	Extended real version of subge0 11497. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
Theorem	xposdif 13005	Extended real version of posdif 11477. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +𝑒 -𝑒𝐴)))
Theorem	xlesubadd 13006	Under certain conditions, the conclusion of lesubadd 11456 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 13007	Lemma for rexmul 13014. (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 13008	Lemma for xmulneg1 13012. (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 13009	Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) = (𝐵 ·e 𝐴))
Theorem	xmul01 13010	Extended real version of mul01 11163. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0)
Theorem	xmul02 13011	Extended real version of mul02 11162. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (0 ·e 𝐴) = 0)
Theorem	xmulneg1 13012	Extended real version of mulneg1 11420. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = -𝑒(𝐴 ·e 𝐵))
Theorem	xmulneg2 13013	Extended real version of mulneg2 11421. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e -𝑒𝐵) = -𝑒(𝐴 ·e 𝐵))
Theorem	rexmul 13014	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 13015	The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ·e :(ℝ* × ℝ*)⟶ℝ*
Theorem	xmulcl 13016	Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) ∈ ℝ*)
Theorem	xmulpnf1 13017	Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e +∞) = +∞)
Theorem	xmulpnf2 13018	Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (+∞ ·e 𝐴) = +∞)
Theorem	xmulmnf1 13019	Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e -∞) = -∞)
Theorem	xmulmnf2 13020	Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (-∞ ·e 𝐴) = -∞)
Theorem	xmulpnf1n 13021	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 13022	Extended real version of mulid1 10982. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 1) = 𝐴)
Theorem	xmulid2 13023	Extended real version of mulid2 10983. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (1 ·e 𝐴) = 𝐴)
Theorem	xmulm1 13024	Extended real version of mulm1 11425. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (-1 ·e 𝐴) = -𝑒𝐴)
Theorem	xmulasslem2 13025	Lemma for xmulass 13030. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((0 < 𝐴 ∧ 𝐴 = -∞) → 𝜑)
Theorem	xmulgt0 13026	Extended real version of mulgt0 11061. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (((𝐴 ∈ ℝ* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 < 𝐵)) → 0 < (𝐴 ·e 𝐵))
Theorem	xmulge0 13027	Extended real version of mulge0 11502. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 ·e 𝐵))
Theorem	xmulasslem 13028*	Lemma for xmulass 13030. (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 13029	Lemma for xmulass 13030. (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 13030	Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 12992 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 13031	Extended real version of lemul1a 11838. (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 13032	Extended real version of lemul2a 11839. (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 13033	Extended real version of lemul1 11836. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+) → (𝐴 ≤ 𝐵 ↔ (𝐴 ·e 𝐶) ≤ (𝐵 ·e 𝐶)))
Theorem	xlemul2 13034	Extended real version of lemul2 11837. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+) → (𝐴 ≤ 𝐵 ↔ (𝐶 ·e 𝐴) ≤ (𝐶 ·e 𝐵)))
Theorem	xltmul1 13035	Extended real version of ltmul1 11834. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐴 ·e 𝐶) < (𝐵 ·e 𝐶)))
Theorem	xltmul2 13036	Extended real version of ltmul2 11835. (Contributed by Mario Carneiro, 8-Sep-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐶 ·e 𝐴) < (𝐶 ·e 𝐵)))
Theorem	xadddilem 13037	Lemma for xadddi 13038. (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 13038	Distributive property for extended real addition and multiplication. Like xaddass 12992, 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 13039	Commuted version of xadddi 13038. (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 13040	The assumption that the multiplier be real in xadddi 13038 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 13041	Commuted version of xadddi2 13040. (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)))
Theorem	x2times 13042	Extended real version of 2times 12118. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (2 ·e 𝐴) = (𝐴 +𝑒 𝐴))
Theorem	xnegcld 13043	Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    ⇒   ⊢ (𝜑 → -𝑒𝐴 ∈ ℝ*)
Theorem	xaddcld 13044	The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (𝐴 +𝑒 𝐵) ∈ ℝ*)
Theorem	xmulcld 13045	Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (𝐴 ·e 𝐵) ∈ ℝ*)
Theorem	xadd4d 13046	Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 11212. (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)))
Theorem	xnn0add4d 13047	Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 13046. (Contributed by AV, 12-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ℕ0*)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ0*)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ0*)    &   ⊢ (𝜑 → 𝐷 ∈ 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
Theorem	xrsupexmnf 13048*	Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.)
⊢ (∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)))
Theorem	xrinfmexpnf 13049*	Adding plus infinity to a set does not affect the existence of its infimum. (Contributed by NM, 19-Jan-2006.)
⊢ (∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦)))
Theorem	xrsupsslem 13050*	Lemma for xrsupss 13052. (Contributed by NM, 25-Oct-2005.)
⊢ ((𝐴 ⊆ ℝ* ∧ (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
Theorem	xrinfmsslem 13051*	Lemma for xrinfmss 13053. (Contributed by NM, 19-Jan-2006.)
⊢ ((𝐴 ⊆ ℝ* ∧ (𝐴 ⊆ ℝ ∨ -∞ ∈ 𝐴)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
Theorem	xrsupss 13052*	Any subset of extended reals has a supremum. (Contributed by NM, 25-Oct-2005.)
⊢ (𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
Theorem	xrinfmss 13053*	Any subset of extended reals has an infimum. (Contributed by NM, 25-Oct-2005.)
⊢ (𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
Theorem	xrinfmss2 13054*	Any subset of extended reals has an infimum. (Contributed by Mario Carneiro, 16-Mar-2014.)
⊢ (𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧)))
Theorem	xrub 13055*	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 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦) ↔ ∀𝑥 ∈ ℝ* (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦)))
Theorem	supxr 13056*	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*) ∧ (∀𝑥 ∈ 𝐴 ¬ 𝐵 < 𝑥 ∧ ∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))) → sup(𝐴, ℝ*, < ) = 𝐵)
Theorem	supxr2 13057*	The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.)
⊢ (((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (∀𝑥 ∈ 𝐴 𝑥 ≤ 𝐵 ∧ ∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦 ∈ 𝐴 𝑥 < 𝑦))) → sup(𝐴, ℝ*, < ) = 𝐵)
Theorem	supxrcl 13058	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*)
Theorem	supxrun 13059	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((𝐴 ∪ 𝐵), ℝ*, < ) = sup(𝐵, ℝ*, < ))
Theorem	supxrmnf 13060	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*, < ))
Theorem	supxrpnf 13061	The supremum of a set of extended reals containing plus infinity is plus infinity. (Contributed by NM, 15-Oct-2005.)
⊢ ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞)
Theorem	supxrunb1 13062*	The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
⊢ (𝐴 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ sup(𝐴, ℝ*, < ) = +∞))
Theorem	supxrunb2 13063*	The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
⊢ (𝐴 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 < 𝑦 ↔ sup(𝐴, ℝ*, < ) = +∞))
Theorem	supxrbnd1 13064*	The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
⊢ (𝐴 ⊆ ℝ* → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ sup(𝐴, ℝ*, < ) < +∞))
Theorem	supxrbnd2 13065*	The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
⊢ (𝐴 ⊆ ℝ* → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ sup(𝐴, ℝ*, < ) < +∞))
Theorem	xrsup0 13066	The supremum of an empty set under the extended reals is minus infinity. (Contributed by NM, 15-Oct-2005.)
⊢ sup(∅, ℝ*, < ) = -∞
Theorem	supxrub 13067	A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by NM, 7-Feb-2006.)
⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → 𝐵 ≤ sup(𝐴, ℝ*, < ))
Theorem	supxrlub 13068*	The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by Mario Carneiro, 13-Sep-2015.)
⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 < sup(𝐴, ℝ*, < ) ↔ ∃𝑥 ∈ 𝐴 𝐵 < 𝑥))
Theorem	supxrleub 13069*	The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by NM, 22-Feb-2006.) (Revised by Mario Carneiro, 6-Sep-2014.)
⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ*) → (sup(𝐴, ℝ*, < ) ≤ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ≤ 𝐵))
Theorem	supxrre 13070*	The real and extended real suprema match when the real supremum exists. (Contributed by NM, 18-Oct-2005.) (Proof shortened by Mario Carneiro, 7-Sep-2014.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ*, < ) = sup(𝐴, ℝ, < ))
Theorem	supxrbnd 13071	The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ*, < ) < +∞) → sup(𝐴, ℝ*, < ) ∈ ℝ)
Theorem	supxrgtmnf 13072	The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → -∞ < sup(𝐴, ℝ*, < ))
Theorem	supxrre1 13073	The supremum of a nonempty set of reals is real iff it is less than plus infinity. (Contributed by NM, 5-Feb-2006.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ sup(𝐴, ℝ*, < ) < +∞))
Theorem	supxrre2 13074	The supremum of a nonempty set of reals is real iff it is not plus infinity. (Contributed by NM, 5-Feb-2006.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ sup(𝐴, ℝ*, < ) ≠ +∞))
Theorem	supxrss 13075	Smaller sets of extended reals have smaller suprema. (Contributed by Mario Carneiro, 1-Apr-2015.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ*) → sup(𝐴, ℝ*, < ) ≤ sup(𝐵, ℝ*, < ))
Theorem	infxrcl 13076	The infimum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 19-Jan-2006.) (Revised by AV, 5-Sep-2020.)
⊢ (𝐴 ⊆ ℝ* → inf(𝐴, ℝ*, < ) ∈ ℝ*)
Theorem	infxrlb 13077	A member of a set of extended reals is greater than or equal to the set's infimum. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 5-Sep-2020.)
⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝐵)
Theorem	infxrgelb 13078*	The infimum of a set of extended reals is greater than or equal to a lower bound. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 5-Sep-2020.)
⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 ≤ inf(𝐴, ℝ*, < ) ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥))
Theorem	infxrre 13079*	The real and extended real infima match when the real infimum exists. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 5-Sep-2020.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → inf(𝐴, ℝ*, < ) = inf(𝐴, ℝ, < ))
Theorem	infxrmnf 13080	The infinimum of a set of extended reals containing minus infinity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Revised by AV, 28-Sep-2020.)
⊢ ((𝐴 ⊆ ℝ* ∧ -∞ ∈ 𝐴) → inf(𝐴, ℝ*, < ) = -∞)
Theorem	xrinf0 13081	The infimum of the empty set under the extended reals is positive infinity. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by AV, 5-Sep-2020.)
⊢ inf(∅, ℝ*, < ) = +∞
Theorem	infxrss 13082	Larger sets of extended reals have smaller infima. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 13-Sep-2020.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ*) → inf(𝐵, ℝ*, < ) ≤ inf(𝐴, ℝ*, < ))
Theorem	reltre 13083*	For all real numbers there is a smaller real number. (Contributed by AV, 5-Sep-2020.)
⊢ ∀𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑦 < 𝑥
Theorem	rpltrp 13084*	For all positive real numbers there is a smaller positive real number. (Contributed by AV, 5-Sep-2020.)
⊢ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ+ 𝑦 < 𝑥
Theorem	reltxrnmnf 13085*	For all extended real numbers not being minus infinity there is a smaller real number. (Contributed by AV, 5-Sep-2020.)
⊢ ∀𝑥 ∈ ℝ* (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)
Theorem	infmremnf 13086	The infimum of the reals is minus infinity. (Contributed by AV, 5-Sep-2020.)
⊢ inf(ℝ, ℝ*, < ) = -∞
Theorem	infmrp1 13087	The infimum of the positive reals is 0. (Contributed by AV, 5-Sep-2020.)
⊢ inf(ℝ+, ℝ, < ) = 0
5.5.4  Real number intervals
Syntax	cioo 13088	Extend class notation with the set of open intervals of extended reals.
class (,)
Syntax	cioc 13089	Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
Syntax	cico 13090	Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
Syntax	cicc 13091	Extend class notation with the set of closed intervals of extended reals.
class [,]
Definition	df-ioo 13092*	Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
Definition	df-ioc 13093*	Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
Definition	df-ico 13094*	Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
Definition	df-icc 13095*	Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
Theorem	ixxval 13096*	Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    ⇒   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})
Theorem	elixx1 13097*	Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    ⇒   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
Theorem	ixxf 13098*	The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    ⇒   ⊢ 𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*
Theorem	ixxex 13099*	The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    ⇒   ⊢ 𝑂 ∈ V
Theorem	ixxssxr 13100*	The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    ⇒   ⊢ (𝐴𝑂𝐵) ⊆ ℝ*