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Theorem List for Metamath Proof Explorer - 12001-12100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmax1 12001 A number is less than or equal to the maximum of it and another. See also max1ALT 12002. (Contributed by NM, 3-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))
 
Theoremmax1ALT 12002 A number is less than or equal to the maximum of it and another. This version of max1 12001 omits the 𝐵 ∈ ℝ antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 12001 is preferred. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 3-Apr-2005.)
(𝐴 ∈ ℝ → 𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))
 
Theoremmax2 12003 A number is less than or equal to the maximum of it and another. (Contributed by NM, 3-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(𝐴𝐵, 𝐵, 𝐴))
 
Theorem2resupmax 12004 The supremum of two real numbers is the maximum of these two numbers. (Contributed by AV, 8-Jun-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ, < ) = if(𝐴𝐵, 𝐵, 𝐴))
 
Theoremmin1 12005 The minimum of two numbers is less than or equal to the first. (Contributed by NM, 3-Aug-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴𝐵, 𝐴, 𝐵) ≤ 𝐴)
 
Theoremmin2 12006 The minimum of two numbers is less than or equal to the second. (Contributed by NM, 3-Aug-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴𝐵, 𝐴, 𝐵) ≤ 𝐵)
 
Theoremmaxle 12007 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by NM, 29-Sep-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (if(𝐴𝐵, 𝐵, 𝐴) ≤ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))
 
Theoremlemin 12008 Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by NM, 3-Aug-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ if(𝐵𝐶, 𝐵, 𝐶) ↔ (𝐴𝐵𝐴𝐶)))
 
Theoremmaxlt 12009 Two ways of saying the maximum of two numbers is less than a third. (Contributed by NM, 3-Aug-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (if(𝐴𝐵, 𝐵, 𝐴) < 𝐶 ↔ (𝐴 < 𝐶𝐵 < 𝐶)))
 
Theoremltmin 12010 Two ways of saying a number is less than the minimum of two others. (Contributed by NM, 1-Sep-2006.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < if(𝐵𝐶, 𝐵, 𝐶) ↔ (𝐴 < 𝐵𝐴 < 𝐶)))
 
Theoremlemaxle 12011 A real number which is less than or equal to a second real number is less than or equal to the maximum/supremum of the second real number and a third real number. (Contributed by AV, 8-Jun-2021.)
(((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ ∧ 𝐴𝐵) → 𝐴 ≤ if(𝐶𝐵, 𝐵, 𝐶))
 
Theoremmax0sub 12012 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)
 
Theoremifle 12013 An if statement transforms an implication into an inequality of terms. (Contributed by Mario Carneiro, 31-Aug-2014.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵𝐴) ∧ (𝜑𝜓)) → if(𝜑, 𝐴, 𝐵) ≤ if(𝜓, 𝐴, 𝐵))
 
Theoremz2ge 12014* There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ∃𝑘 ∈ ℤ (𝑀𝑘𝑁𝑘))
 
Theoremqbtwnre 12015* The rational numbers are dense in : 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 < 𝑥𝑥 < 𝐵))
 
Theoremqbtwnxr 12016* The rational numbers are dense in *: 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 < 𝑥𝑥 < 𝐵))
 
Theoremqsqueeze 12017* 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 < 𝑥𝐴 < 𝑥)) → 𝐴 = 0)
 
Theoremqextltlem 12018* Lemma for qextlt 12019 and qextle . (Contributed by Mario Carneiro, 3-Oct-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐴𝑥 < 𝐵) ∧ ¬ (𝑥𝐴𝑥𝐵))))
 
Theoremqextlt 12019* An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℚ (𝑥 < 𝐴𝑥 < 𝐵)))
 
Theoremqextle 12020* An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℚ (𝑥𝐴𝑥𝐵)))
 
Theoremxralrple 12021* 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) → (𝐴𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + 𝑥)))
 
Theoremalrple 12022* 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) → (𝐴𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + 𝑥)))
 
Theoremxnegeq 12023 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)
 
Theoremxnegex 12024 A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)
-𝑒𝐴 ∈ V
 
Theoremxnegpnf 12025 Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
-𝑒+∞ = -∞
 
Theoremxnegmnf 12026 Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
-𝑒-∞ = +∞
 
Theoremrexneg 12027 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)
 
Theoremxneg0 12028 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
-𝑒0 = 0
 
Theoremxnegcl 12029 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*)
 
Theoremxnegneg 12030 Extended real version of negneg 10316. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴)
 
Theoremxneg11 12031 Extended real version of neg11 10317. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵𝐴 = 𝐵))
 
Theoremxltnegi 12032 Forward direction of xltneg 12033. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴)
 
Theoremxltneg 12033 Extended real version of ltneg 10513. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴))
 
Theoremxleneg 12034 Extended real version of leneg 10516. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴))
 
Theoremxlt0neg1 12035 Extended real version of lt0neg1 10519. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴))
 
Theoremxlt0neg2 12036 Extended real version of lt0neg2 10520. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0))
 
Theoremxle0neg1 12037 Extended real version of le0neg1 10521. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴))
 
Theoremxle0neg2 12038 Extended real version of le0neg2 10522. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (0 ≤ 𝐴 ↔ -𝑒𝐴 ≤ 0))
 
Theoremxaddval 12039 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 12040 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
+𝑒 :(ℝ* × ℝ*)⟶ℝ*
 
Theoremxmulval 12041 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 12042 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)
 
Theoremxaddpnf2 12043 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞)
 
Theoremxaddmnf1 12044 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)
 
Theoremxaddmnf2 12045 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞)
 
Theorempnfaddmnf 12046 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 12047 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 12048 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 12049 Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 -𝑒𝐵) = (𝐴𝐵))
 
Theoremrexaddd 12050 The extended real addition operation when both arguments are real. Deduction version of rexadd 12048. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))
 
Theoremxnn0xaddcl 12051 The extended nonnegative integers are closed under extended addition. (Contributed by AV, 10-Dec-2020.)
((𝐴 ∈ ℕ0*𝐵 ∈ ℕ0*) → (𝐴 +𝑒 𝐵) ∈ ℕ0*)
 
Theoremxaddnemnf 12052 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 12053 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 12054 Extended real version of negid 10313. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0)
 
Theoremxaddcl 12055 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 12056 The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = (𝐵 +𝑒 𝐴))
 
Theoremxaddid1 12057 Extended real version of addid1 10201. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴)
 
Theoremxaddid2 12058 Extended real version of addid2 10204. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴)
 
Theoremxaddid1d 12059 0 is a right identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → (𝐴 +𝑒 0) = 𝐴)
 
Theoremxnn0lenn0nn0 12060 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 12061 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 12062 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 12063 Extended real version of xnegdi 12063. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 𝐵) = (-𝑒𝐴 +𝑒 -𝑒𝐵))
 
Theoremxaddass 12064 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 12065, 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 12065 Associativity of extended real addition. See xaddass 12064 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 12066 Extended real version of pncan 10272. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → ((𝐴 +𝑒 𝐵) +𝑒 -𝑒𝐵) = 𝐴)
 
Theoremxnpcan 12067 Extended real version of npcan 10275. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)
 
Theoremxleadd1a 12068 Extended real version of leadd1 10481; 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 12069 Commuted form of xleadd1a 12068. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ 𝐴𝐵) → (𝐶 +𝑒 𝐴) ≤ (𝐶 +𝑒 𝐵))
 
Theoremxleadd1 12070 Weakened version of xleadd1a 12068 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 12071 Extended real version of ltadd1 10480. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 +𝑒 𝐶) < (𝐵 +𝑒 𝐶)))
 
Theoremxltadd2 12072 Extended real version of ltadd2 10126. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 +𝑒 𝐴) < (𝐶 +𝑒 𝐵)))
 
Theoremxaddge0 12073 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 12074 Extended real version of le2add 10495. (Contributed by Mario Carneiro, 23-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴𝐶𝐵𝐷) → (𝐴 +𝑒 𝐵) ≤ (𝐶 +𝑒 𝐷)))
 
Theoremxlt2add 12075 Extended real version of lt2add 10498. Note that ltleadd 10496, 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 12076 Extended real version of subge0 10526. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵𝐴))
 
Theoremxposdif 12077 Extended real version of posdif 10506. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +𝑒 -𝑒𝐴)))
 
Theoremxlesubadd 12078 Under certain conditions, the conclusion of lesubadd 10485 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 12079 Lemma for rexmul 12086. (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 12080 Lemma for xmulneg1 12084. (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 12081 Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) = (𝐵 ·e 𝐴))
 
Theoremxmul01 12082 Extended real version of mul01 10200. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0)
 
Theoremxmul02 12083 Extended real version of mul02 10199. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 ·e 𝐴) = 0)
 
Theoremxmulneg1 12084 Extended real version of mulneg1 10451. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = -𝑒(𝐴 ·e 𝐵))
 
Theoremxmulneg2 12085 Extended real version of mulneg2 10452. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e -𝑒𝐵) = -𝑒(𝐴 ·e 𝐵))
 
Theoremrexmul 12086 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 12087 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
·e :(ℝ* × ℝ*)⟶ℝ*
 
Theoremxmulcl 12088 Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) ∈ ℝ*)
 
Theoremxmulpnf1 12089 Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e +∞) = +∞)
 
Theoremxmulpnf2 12090 Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (+∞ ·e 𝐴) = +∞)
 
Theoremxmulmnf1 12091 Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e -∞) = -∞)
 
Theoremxmulmnf2 12092 Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (-∞ ·e 𝐴) = -∞)
 
Theoremxmulpnf1n 12093 Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 < 0) → (𝐴 ·e +∞) = -∞)
 
Theoremxmulid1 12094 Extended real version of mulid1 10022. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 ·e 1) = 𝐴)
 
Theoremxmulid2 12095 Extended real version of mulid2 10023. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (1 ·e 𝐴) = 𝐴)
 
Theoremxmulm1 12096 Extended real version of mulm1 10456. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (-1 ·e 𝐴) = -𝑒𝐴)
 
Theoremxmulasslem2 12097 Lemma for xmulass 12102. (Contributed by Mario Carneiro, 20-Aug-2015.)
((0 < 𝐴𝐴 = -∞) → 𝜑)
 
Theoremxmulgt0 12098 Extended real version of mulgt0 10100. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 < 𝐵)) → 0 < (𝐴 ·e 𝐵))
 
Theoremxmulge0 12099 Extended real version of mulge0 10531. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 ·e 𝐵))
 
Theoremxmulasslem 12100* Lemma for xmulass 12102. (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)
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