P. 133 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13201-13300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xrlemin 13201	Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶)))
Theorem	max1 13202	A number is less than or equal to the maximum of it and another. See also max1ALT 13203. (Contributed by NM, 3-Apr-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
Theorem	max1ALT 13203	A number is less than or equal to the maximum of it and another. This version of max1 13202 omits the 𝐵 ∈ ℝ antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 13202 is preferred. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 3-Apr-2005.)
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
Theorem	max2 13204	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(𝐴 ≤ 𝐵, 𝐵, 𝐴))
Theorem	2resupmax 13205	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(𝐴 ≤ 𝐵, 𝐵, 𝐴))
Theorem	min1 13206	The minimum of two numbers is less than or equal to the first. (Contributed by NM, 3-Aug-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴)
Theorem	min2 13207	The minimum of two numbers is less than or equal to the second. (Contributed by NM, 3-Aug-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵)
Theorem	maxle 13208	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(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶)))
Theorem	lemin 13209	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(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶)))
Theorem	maxlt 13210	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(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶 ↔ (𝐴 < 𝐶 ∧ 𝐵 < 𝐶)))
Theorem	ltmin 13211	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(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))
Theorem	lemaxle 13212	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 ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶))
Theorem	max0sub 13213	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 13214	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 13215*	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 13216*	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 13217*	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 13218*	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 13219*	Lemma for qextlt 13220 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 13220*	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 13221*	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 13222*	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 13223*	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 13224	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 13225	A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ -𝑒𝐴 ∈ V
Theorem	xnegpnf 13226	Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
⊢ -𝑒+∞ = -∞
Theorem	xnegmnf 13227	Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
⊢ -𝑒-∞ = +∞
Theorem	rexneg 13228	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 13229	The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ -𝑒0 = 0
Theorem	xnegcl 13230	Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*)
Theorem	xnegneg 13231	Extended real version of negneg 11496. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴)
Theorem	xneg11 13232	Extended real version of neg11 11497. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵 ↔ 𝐴 = 𝐵))
Theorem	xltnegi 13233	Forward direction of xltneg 13234. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴)
Theorem	xltneg 13234	Extended real version of ltneg 11702. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴))
Theorem	xleneg 13235	Extended real version of leneg 11705. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴))
Theorem	xlt0neg1 13236	Extended real version of lt0neg1 11708. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴))
Theorem	xlt0neg2 13237	Extended real version of lt0neg2 11709. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0))
Theorem	xle0neg1 13238	Extended real version of le0neg1 11710. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴))
Theorem	xle0neg2 13239	Extended real version of le0neg2 11711. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (𝐴 ∈ ℝ* → (0 ≤ 𝐴 ↔ -𝑒𝐴 ≤ 0))
Theorem	xaddval 13240	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 13241	The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ*
Theorem	xmulval 13242	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 13243	Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)
Theorem	xaddpnf2 13244	Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞)
Theorem	xaddmnf1 13245	Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)
Theorem	xaddmnf2 13246	Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞)
Theorem	pnfaddmnf 13247	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 13248	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 13249	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 13250	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 13251	The extended real addition operation when both arguments are real. Deduction version of rexadd 13249. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))
Theorem	xnn0xaddcl 13252	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 13253	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 13254	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 13255	Extended real version of negid 11493. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0)
Theorem	xaddcl 13256	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 13257	The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = (𝐵 +𝑒 𝐴))
Theorem	xaddrid 13258	Extended real version of addrid 11378. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴)
Theorem	xaddlid 13259	Extended real version of addlid 11381. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴)
Theorem	xaddridd 13260	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	xnn0lem1lt 13261	Extended nonnegative integer ordering relation. (Contributed by Thierry Arnoux, 30-Jul-2023.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁))
Theorem	xnn0lenn0nn0 13262	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 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℕ0)
Theorem	xnn0le2is012 13263	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))
Theorem	xnn0xadd0 13264	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 13265	Extended real version of negdi 11503. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 𝐵) = (-𝑒𝐴 +𝑒 -𝑒𝐵))
Theorem	xaddass 13266	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 13267, 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 13267	Associativity of extended real addition. See xaddass 13266 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 13268	Extended real version of pncan 11451. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 𝐵) +𝑒 -𝑒𝐵) = 𝐴)
Theorem	xnpcan 13269	Extended real version of npcan 11454. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)
Theorem	xleadd1a 13270	Extended real version of leadd1 11670; 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 13271	Commuted form of xleadd1a 13270. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐶 +𝑒 𝐴) ≤ (𝐶 +𝑒 𝐵))
Theorem	xleadd1 13272	Weakened version of xleadd1a 13270 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 13273	Extended real version of ltadd1 11669. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 +𝑒 𝐶) < (𝐵 +𝑒 𝐶)))
Theorem	xltadd2 13274	Extended real version of ltadd2 11302. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 +𝑒 𝐴) < (𝐶 +𝑒 𝐵)))
Theorem	xaddge0 13275	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 13276	Extended real version of le2add 11684. (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 13277	Extended real version of lt2add 11687. Note that ltleadd 11685, 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 13278	Extended real version of subge0 11715. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
Theorem	xposdif 13279	Extended real version of posdif 11695. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +𝑒 -𝑒𝐴)))
Theorem	xlesubadd 13280	Under certain conditions, the conclusion of lesubadd 11674 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 13281	Lemma for rexmul 13288. (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 13282	Lemma for xmulneg1 13286. (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 13283	Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) = (𝐵 ·e 𝐴))
Theorem	xmul01 13284	Extended real version of mul01 11377. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0)
Theorem	xmul02 13285	Extended real version of mul02 11376. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (0 ·e 𝐴) = 0)
Theorem	xmulneg1 13286	Extended real version of mulneg1 11638. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = -𝑒(𝐴 ·e 𝐵))
Theorem	xmulneg2 13287	Extended real version of mulneg2 11639. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e -𝑒𝐵) = -𝑒(𝐴 ·e 𝐵))
Theorem	rexmul 13288	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 13289	The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ·e :(ℝ* × ℝ*)⟶ℝ*
Theorem	xmulcl 13290	Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) ∈ ℝ*)
Theorem	xmulpnf1 13291	Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e +∞) = +∞)
Theorem	xmulpnf2 13292	Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (+∞ ·e 𝐴) = +∞)
Theorem	xmulmnf1 13293	Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e -∞) = -∞)
Theorem	xmulmnf2 13294	Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (-∞ ·e 𝐴) = -∞)
Theorem	xmulpnf1n 13295	Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (𝐴 ·e +∞) = -∞)
Theorem	xmulrid 13296	Extended real version of mulrid 11194. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 1) = 𝐴)
Theorem	xmullid 13297	Extended real version of mullid 11195. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (1 ·e 𝐴) = 𝐴)
Theorem	xmulm1 13298	Extended real version of mulm1 11643. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (-1 ·e 𝐴) = -𝑒𝐴)
Theorem	xmulasslem2 13299	Lemma for xmulass 13304. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((0 < 𝐴 ∧ 𝐴 = -∞) → 𝜑)
Theorem	xmulgt0 13300	Extended real version of mulgt0 11275. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (((𝐴 ∈ ℝ* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 < 𝐵)) → 0 < (𝐴 ·e 𝐵))