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Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrexadd 12101 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 12102 Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 -𝑒𝐵) = (𝐴𝐵))
 
Theoremrexaddd 12103 The extended real addition operation when both arguments are real. Deduction version of rexadd 12101. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))
 
Theoremxnn0xaddcl 12104 The extended nonnegative integers are closed under extended addition. (Contributed by AV, 10-Dec-2020.)
((𝐴 ∈ ℕ0*𝐵 ∈ ℕ0*) → (𝐴 +𝑒 𝐵) ∈ ℕ0*)
 
Theoremxaddnemnf 12105 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 12106 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 12107 Extended real version of negid 10366. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0)
 
Theoremxaddcl 12108 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 12109 The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = (𝐵 +𝑒 𝐴))
 
Theoremxaddid1 12110 Extended real version of addid1 10254. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴)
 
Theoremxaddid2 12111 Extended real version of addid2 10257. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴)
 
Theoremxaddid1d 12112 0 is a right identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → (𝐴 +𝑒 0) = 𝐴)
 
Theoremxnn0lenn0nn0 12113 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 12114 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 12115 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 12116 Extended real version of xnegdi 12116. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 𝐵) = (-𝑒𝐴 +𝑒 -𝑒𝐵))
 
Theoremxaddass 12117 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 12118, 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 12118 Associativity of extended real addition. See xaddass 12117 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 12119 Extended real version of pncan 10325. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → ((𝐴 +𝑒 𝐵) +𝑒 -𝑒𝐵) = 𝐴)
 
Theoremxnpcan 12120 Extended real version of npcan 10328. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)
 
Theoremxleadd1a 12121 Extended real version of leadd1 10534; 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 12122 Commuted form of xleadd1a 12121. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ 𝐴𝐵) → (𝐶 +𝑒 𝐴) ≤ (𝐶 +𝑒 𝐵))
 
Theoremxleadd1 12123 Weakened version of xleadd1a 12121 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 12124 Extended real version of ltadd1 10533. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 +𝑒 𝐶) < (𝐵 +𝑒 𝐶)))
 
Theoremxltadd2 12125 Extended real version of ltadd2 10179. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 +𝑒 𝐴) < (𝐶 +𝑒 𝐵)))
 
Theoremxaddge0 12126 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 12127 Extended real version of le2add 10548. (Contributed by Mario Carneiro, 23-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴𝐶𝐵𝐷) → (𝐴 +𝑒 𝐵) ≤ (𝐶 +𝑒 𝐷)))
 
Theoremxlt2add 12128 Extended real version of lt2add 10551. Note that ltleadd 10549, 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 12129 Extended real version of subge0 10579. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵𝐴))
 
Theoremxposdif 12130 Extended real version of posdif 10559. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +𝑒 -𝑒𝐴)))
 
Theoremxlesubadd 12131 Under certain conditions, the conclusion of lesubadd 10538 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 12132 Lemma for rexmul 12139. (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 12133 Lemma for xmulneg1 12137. (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 12134 Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) = (𝐵 ·e 𝐴))
 
Theoremxmul01 12135 Extended real version of mul01 10253. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0)
 
Theoremxmul02 12136 Extended real version of mul02 10252. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 ·e 𝐴) = 0)
 
Theoremxmulneg1 12137 Extended real version of mulneg1 10504. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = -𝑒(𝐴 ·e 𝐵))
 
Theoremxmulneg2 12138 Extended real version of mulneg2 10505. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e -𝑒𝐵) = -𝑒(𝐴 ·e 𝐵))
 
Theoremrexmul 12139 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 12140 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
·e :(ℝ* × ℝ*)⟶ℝ*
 
Theoremxmulcl 12141 Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) ∈ ℝ*)
 
Theoremxmulpnf1 12142 Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e +∞) = +∞)
 
Theoremxmulpnf2 12143 Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (+∞ ·e 𝐴) = +∞)
 
Theoremxmulmnf1 12144 Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e -∞) = -∞)
 
Theoremxmulmnf2 12145 Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (-∞ ·e 𝐴) = -∞)
 
Theoremxmulpnf1n 12146 Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 < 0) → (𝐴 ·e +∞) = -∞)
 
Theoremxmulid1 12147 Extended real version of mulid1 10075. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 ·e 1) = 𝐴)
 
Theoremxmulid2 12148 Extended real version of mulid2 10076. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (1 ·e 𝐴) = 𝐴)
 
Theoremxmulm1 12149 Extended real version of mulm1 10509. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (-1 ·e 𝐴) = -𝑒𝐴)
 
Theoremxmulasslem2 12150 Lemma for xmulass 12155. (Contributed by Mario Carneiro, 20-Aug-2015.)
((0 < 𝐴𝐴 = -∞) → 𝜑)
 
Theoremxmulgt0 12151 Extended real version of mulgt0 10153. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 < 𝐵)) → 0 < (𝐴 ·e 𝐵))
 
Theoremxmulge0 12152 Extended real version of mulge0 10584. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 ·e 𝐵))
 
Theoremxmulasslem 12153* Lemma for xmulass 12155. (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 12154 Lemma for xmulass 12155. (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 12155 Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 12117 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 12156 Extended real version of lemul1a 10915. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶)) ∧ 𝐴𝐵) → (𝐴 ·e 𝐶) ≤ (𝐵 ·e 𝐶))
 
Theoremxlemul2a 12157 Extended real version of lemul2a 10916. (Contributed by Mario Carneiro, 8-Sep-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶)) ∧ 𝐴𝐵) → (𝐶 ·e 𝐴) ≤ (𝐶 ·e 𝐵))
 
Theoremxlemul1 12158 Extended real version of lemul1 10913. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ+) → (𝐴𝐵 ↔ (𝐴 ·e 𝐶) ≤ (𝐵 ·e 𝐶)))
 
Theoremxlemul2 12159 Extended real version of lemul2 10914. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ+) → (𝐴𝐵 ↔ (𝐶 ·e 𝐴) ≤ (𝐶 ·e 𝐵)))
 
Theoremxltmul1 12160 Extended real version of ltmul1 10911. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐴 ·e 𝐶) < (𝐵 ·e 𝐶)))
 
Theoremxltmul2 12161 Extended real version of ltmul2 10912. (Contributed by Mario Carneiro, 8-Sep-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐶 ·e 𝐴) < (𝐶 ·e 𝐵)))
 
Theoremxadddilem 12162 Lemma for xadddi 12163. (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 12163 Distributive property for extended real addition and multiplication. Like xaddass 12117, 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 12164 Commuted version of xadddi 12163. (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 12165 The assumption that the multiplier be real in xadddi 12163 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 12166 Commuted version of xadddi2 12165. (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 12167 Extended real version of 2times 11183. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (2 ·e 𝐴) = (𝐴 +𝑒 𝐴))
 
Theoremxnegcld 12168 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → -𝑒𝐴 ∈ ℝ*)
 
Theoremxaddcld 12169 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 12170 Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐴 ·e 𝐵) ∈ ℝ*)
 
Theoremxadd4d 12171 Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 10302. (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 12172 Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 12171. (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 12173* 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 12174* 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 12175* Lemma for xrsupss 12177. (Contributed by NM, 25-Oct-2005.)
((𝐴 ⊆ ℝ* ∧ (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴)) → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
 
Theoremxrinfmsslem 12176* Lemma for xrinfmss 12178. (Contributed by NM, 19-Jan-2006.)
((𝐴 ⊆ ℝ* ∧ (𝐴 ⊆ ℝ ∨ -∞ ∈ 𝐴)) → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
 
Theoremxrsupss 12177* 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 12178* 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 12179* 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 12180* 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 12181* 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 12182* 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 12183 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 12184 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 12185 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*, < ))
 
Theoremsupxrpnf 12186 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*, < ) = +∞)
 
Theoremsupxrunb1 12187* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
(𝐴 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑦𝐴 𝑥𝑦 ↔ sup(𝐴, ℝ*, < ) = +∞))
 
Theoremsupxrunb2 12188* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
(𝐴 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑦𝐴 𝑥 < 𝑦 ↔ sup(𝐴, ℝ*, < ) = +∞))
 
Theoremsupxrbnd1 12189* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
(𝐴 ⊆ ℝ* → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 ↔ sup(𝐴, ℝ*, < ) < +∞))
 
Theoremsupxrbnd2 12190* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
(𝐴 ⊆ ℝ* → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥 ↔ sup(𝐴, ℝ*, < ) < +∞))
 
Theoremxrsup0 12191 The supremum of an empty set under the extended reals is minus infinity. (Contributed by NM, 15-Oct-2005.)
sup(∅, ℝ*, < ) = -∞
 
Theoremsupxrub 12192 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*𝐵𝐴) → 𝐵 ≤ sup(𝐴, ℝ*, < ))
 
Theoremsupxrlub 12193* 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*, < ) ↔ ∃𝑥𝐴 𝐵 < 𝑥))
 
Theoremsupxrleub 12194* 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 ↔ ∀𝑥𝐴 𝑥𝐵))
 
Theoremsupxrre 12195* 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 ∀𝑦𝐴 𝑦𝑥) → sup(𝐴, ℝ*, < ) = sup(𝐴, ℝ, < ))
 
Theoremsupxrbnd 12196 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)
 
Theoremsupxrgtmnf 12197 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*, < ))
 
Theoremsupxrre1 12198 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*, < ) < +∞))
 
Theoremsupxrre2 12199 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*, < ) ≠ +∞))
 
Theoremsupxrss 12200 Smaller sets of extended reals have smaller suprema. (Contributed by Mario Carneiro, 1-Apr-2015.)
((𝐴𝐵𝐵 ⊆ ℝ*) → sup(𝐴, ℝ*, < ) ≤ sup(𝐵, ℝ*, < ))
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