P. 100 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9901-10000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xrlttrd 9901	Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 < 𝐶)
Theorem	xrlelttrd 9902	Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 < 𝐶)
Theorem	xrltletrd 9903	Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 < 𝐶)
Theorem	xrletrd 9904	Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐶)
Theorem	xrltne 9905	'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴)
Theorem	nltpnft 9906	An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
Theorem	npnflt 9907	An extended real is less than plus infinity iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
⊢ (𝐴 ∈ ℝ* → (𝐴 < +∞ ↔ 𝐴 ≠ +∞))
Theorem	xgepnf 9908	An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
⊢ (𝐴 ∈ ℝ* → (+∞ ≤ 𝐴 ↔ 𝐴 = +∞))
Theorem	ngtmnft 9909	An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)
⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))
Theorem	nmnfgt 9910	An extended real is greater than minus infinite iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
⊢ (𝐴 ∈ ℝ* → (-∞ < 𝐴 ↔ 𝐴 ≠ -∞))
Theorem	xrrebnd 9911	An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)
⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
Theorem	xrre 9912	A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)
Theorem	xrre2 9913	An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ)
Theorem	xrre3 9914	A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → 𝐴 ∈ ℝ)
Theorem	ge0gtmnf 9915	A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)
Theorem	ge0nemnf 9916	A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞)
Theorem	xrrege0 9917	A nonnegative extended real that is less than a real bound is real. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)
Theorem	z2ge 9918*	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	xnegeq 9919	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	xnegpnf 9920	Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
⊢ -𝑒+∞ = -∞
Theorem	xnegmnf 9921	Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
⊢ -𝑒-∞ = +∞
Theorem	rexneg 9922	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 9923	The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ -𝑒0 = 0
Theorem	xnegcl 9924	Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*)
Theorem	xnegneg 9925	Extended real version of negneg 8293. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴)
Theorem	xneg11 9926	Extended real version of neg11 8294. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵 ↔ 𝐴 = 𝐵))
Theorem	xltnegi 9927	Forward direction of xltneg 9928. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴)
Theorem	xltneg 9928	Extended real version of ltneg 8506. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴))
Theorem	xleneg 9929	Extended real version of leneg 8509. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴))
Theorem	xlt0neg1 9930	Extended real version of lt0neg1 8512. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴))
Theorem	xlt0neg2 9931	Extended real version of lt0neg2 8513. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0))
Theorem	xle0neg1 9932	Extended real version of le0neg1 8514. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴))
Theorem	xle0neg2 9933	Extended real version of le0neg2 8515. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (𝐴 ∈ ℝ* → (0 ≤ 𝐴 ↔ -𝑒𝐴 ≤ 0))
Theorem	xrpnfdc 9934	An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = +∞)
Theorem	xrmnfdc 9935	An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = -∞)
Theorem	xaddf 9936	The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ*
Theorem	xaddval 9937	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	xaddpnf1 9938	Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)
Theorem	xaddpnf2 9939	Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞)
Theorem	xaddmnf1 9940	Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)
Theorem	xaddmnf2 9941	Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞)
Theorem	pnfaddmnf 9942	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 9943	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 9944	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 9945	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 9946	The extended real addition operation when both arguments are real. Deduction version of rexadd 9944. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))
Theorem	xnegcld 9947	Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    ⇒   ⊢ (𝜑 → -𝑒𝐴 ∈ ℝ*)
Theorem	xrex 9948	The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
⊢ ℝ* ∈ V
Theorem	xaddnemnf 9949	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 9950	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 9951	Extended real version of negid 8290. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0)
Theorem	xaddcl 9952	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 9953	The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = (𝐵 +𝑒 𝐴))
Theorem	xaddid1 9954	Extended real version of addrid 8181. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴)
Theorem	xaddid2 9955	Extended real version of addlid 8182. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴)
Theorem	xaddid1d 9956	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	xnn0lenn0nn0 9957	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 9958	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 9959	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 9960	Extended real version of negdi 8300. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 𝐵) = (-𝑒𝐴 +𝑒 -𝑒𝐵))
Theorem	xaddass 9961	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 9962, 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 9962	Associativity of extended real addition. See xaddass 9961 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 9963	Extended real version of pncan 8249. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 𝐵) +𝑒 -𝑒𝐵) = 𝐴)
Theorem	xnpcan 9964	Extended real version of npcan 8252. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)
Theorem	xleadd1a 9965	Extended real version of leadd1 8474; 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 9966	Commuted form of xleadd1a 9965. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐶 +𝑒 𝐴) ≤ (𝐶 +𝑒 𝐵))
Theorem	xleadd1 9967	Weakened version of xleadd1a 9965 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 9968	Extended real version of ltadd1 8473. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Jim Kingdon, 16-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 +𝑒 𝐶) < (𝐵 +𝑒 𝐶)))
Theorem	xltadd2 9969	Extended real version of ltadd2 8463. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 +𝑒 𝐴) < (𝐶 +𝑒 𝐵)))
Theorem	xaddge0 9970	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 9971	Extended real version of le2add 8488. (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 9972	Extended real version of lt2add 8489. Note that ltleadd 8490, 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 9973	Extended real version of subge0 8519. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
Theorem	xposdif 9974	Extended real version of posdif 8499. (Contributed by Mario Carneiro, 24-Aug-2015.) (Revised by Jim Kingdon, 17-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +𝑒 -𝑒𝐴)))
Theorem	xlesubadd 9975	Under certain conditions, the conclusion of lesubadd 8478 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	xaddcld 9976	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	xadd4d 9977	Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 8212. (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 9978	Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 9977. (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)))
Theorem	xleaddadd 9979	Cancelling a factor of two in ≤ (expressed as addition rather than as a factor to avoid extended real multiplication). (Contributed by Jim Kingdon, 18-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ (𝐴 +𝑒 𝐴) ≤ (𝐵 +𝑒 𝐵)))
4.5.3  Real number intervals
Syntax	cioo 9980	Extend class notation with the set of open intervals of extended reals.
class (,)
Syntax	cioc 9981	Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
Syntax	cico 9982	Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
Syntax	cicc 9983	Extend class notation with the set of closed intervals of extended reals.
class [,]
Definition	df-ioo 9984*	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 9985*	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 9986*	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 9987*	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 9988*	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 9989*	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 9990*	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 9991*	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 9992*	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*
Theorem	elixx3g 9993*	Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show 𝐴 ∈ ℝ* and 𝐵 ∈ ℝ*. (Contributed by Mario Carneiro, 3-Nov-2013.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    ⇒   ⊢ (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
Theorem	ixxssixx 9994*	An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑈𝑦)})    &   ⊢ ((𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐴𝑅𝑤 → 𝐴𝑇𝑤))    &   ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵))    ⇒   ⊢ (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵)
Theorem	ixxdisj 9995*	Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑈𝑦)})    &   ⊢ ((𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐵𝑇𝑤 ↔ ¬ 𝑤𝑆𝐵))    ⇒   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴𝑂𝐵) ∩ (𝐵𝑃𝐶)) = ∅)
Theorem	ixxss1 9996*	Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑆𝑦)})    &   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤))    ⇒   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶))
Theorem	ixxss2 9997*	Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑇𝑦)})    &   ⊢ ((𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝑤𝑇𝐵 ∧ 𝐵𝑊𝐶) → 𝑤𝑆𝐶))    ⇒   ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵𝑊𝐶) → (𝐴𝑃𝐵) ⊆ (𝐴𝑂𝐶))
Theorem	ixxss12 9998*	Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})    &   ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑈𝑦)})    &   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴𝑊𝐶 ∧ 𝐶𝑇𝑤) → 𝐴𝑅𝑤))    &   ⊢ ((𝑤 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤𝑈𝐷 ∧ 𝐷𝑋𝐵) → 𝑤𝑆𝐵))    ⇒   ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴𝑊𝐶 ∧ 𝐷𝑋𝐵)) → (𝐶𝑃𝐷) ⊆ (𝐴𝑂𝐵))
Theorem	iooex 9999	The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ (,) ∈ V
Theorem	iooval 10000*	Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})