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Theorem List for Intuitionistic Logic Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmnfltxr 9601 Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)
((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)

Theorempnfnlt 9602 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
(𝐴 ∈ ℝ* → ¬ +∞ < 𝐴)

Theoremnltmnf 9603 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
(𝐴 ∈ ℝ* → ¬ 𝐴 < -∞)

Theorempnfge 9604 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
(𝐴 ∈ ℝ*𝐴 ≤ +∞)

Theorem0lepnf 9605 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≤ +∞

Theoremnn0pnfge0 9606 If a number is a nonnegative integer or positive infinity, it is greater than or equal to 0. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
((𝑁 ∈ ℕ0𝑁 = +∞) → 0 ≤ 𝑁)

Theoremmnfle 9607 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
(𝐴 ∈ ℝ* → -∞ ≤ 𝐴)

Theoremxrltnsym 9608 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))

Theoremxrltnsym2 9609 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ¬ (𝐴 < 𝐵𝐵 < 𝐴))

Theoremxrlttr 9610 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))

Theoremxrltso 9611 'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
< Or ℝ*

Theoremxrlttri3 9612 Extended real version of lttri3 7867. (Contributed by NM, 9-Feb-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))

Theoremxrltle 9613 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵𝐴𝐵))

Theoremxrltled 9614 'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle 9613. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴𝐵)

Theoremxrleid 9615 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
(𝐴 ∈ ℝ*𝐴𝐴)

Theoremxrleidd 9616 'Less than or equal to' is reflexive for extended reals. Deduction form of xrleid 9615. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)       (𝜑𝐴𝐴)

Theoremxrletri3 9617 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴)))

Theoremxrlelttr 9618 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))

Theoremxrltletr 9619 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 < 𝐵𝐵𝐶) → 𝐴 < 𝐶))

Theoremxrletr 9620 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))

Theoremxrlttrd 9621 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐵 < 𝐶)       (𝜑𝐴 < 𝐶)

Theoremxrlelttrd 9622 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 < 𝐶)       (𝜑𝐴 < 𝐶)

Theoremxrltletrd 9623 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴 < 𝐶)

Theoremxrletrd 9624 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremxrltne 9625 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → 𝐵𝐴)

Theoremnltpnft 9626 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
(𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))

Theoremnpnflt 9627 An extended real is less than plus infinity iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
(𝐴 ∈ ℝ* → (𝐴 < +∞ ↔ 𝐴 ≠ +∞))

Theoremxgepnf 9628 An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ* → (+∞ ≤ 𝐴𝐴 = +∞))

Theoremngtmnft 9629 An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)
(𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))

Theoremnmnfgt 9630 An extended real is greater than minus infinite iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
(𝐴 ∈ ℝ* → (-∞ < 𝐴𝐴 ≠ -∞))

Theoremxrrebnd 9631 An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)
(𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴𝐴 < +∞)))

Theoremxrre 9632 A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ) ∧ (-∞ < 𝐴𝐴𝐵)) → 𝐴 ∈ ℝ)

Theoremxrre2 9633 An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵𝐵 < 𝐶)) → 𝐵 ∈ ℝ)

Theoremxrre3 9634 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ) ∧ (𝐵𝐴𝐴 < +∞)) → 𝐴 ∈ ℝ)

Theoremge0gtmnf 9635 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)

Theoremge0nemnf 9636 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞)

Theoremxrrege0 9637 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 ≤ 𝐴𝐴𝐵)) → 𝐴 ∈ ℝ)

Theoremz2ge 9638* 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))

Theoremxnegeq 9639 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)

Theoremxnegpnf 9640 Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
-𝑒+∞ = -∞

Theoremxnegmnf 9641 Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
-𝑒-∞ = +∞

Theoremrexneg 9642 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 9643 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
-𝑒0 = 0

Theoremxnegcl 9644 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*)

Theoremxnegneg 9645 Extended real version of negneg 8035. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴)

Theoremxneg11 9646 Extended real version of neg11 8036. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵𝐴 = 𝐵))

Theoremxltnegi 9647 Forward direction of xltneg 9648. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴)

Theoremxltneg 9648 Extended real version of ltneg 8247. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴))

Theoremxleneg 9649 Extended real version of leneg 8250. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴))

Theoremxlt0neg1 9650 Extended real version of lt0neg1 8253. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴))

Theoremxlt0neg2 9651 Extended real version of lt0neg2 8254. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0))

Theoremxle0neg1 9652 Extended real version of le0neg1 8255. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴))

Theoremxle0neg2 9653 Extended real version of le0neg2 8256. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (0 ≤ 𝐴 ↔ -𝑒𝐴 ≤ 0))

Theoremxrpnfdc 9654 An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
(𝐴 ∈ ℝ*DECID 𝐴 = +∞)

Theoremxrmnfdc 9655 An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
(𝐴 ∈ ℝ*DECID 𝐴 = -∞)

Theoremxaddf 9656 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
+𝑒 :(ℝ* × ℝ*)⟶ℝ*

Theoremxaddval 9657 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))))))

Theoremxaddpnf1 9658 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)

Theoremxaddpnf2 9659 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞)

Theoremxaddmnf1 9660 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)

Theoremxaddmnf2 9661 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞)

Theorempnfaddmnf 9662 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 9663 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 9664 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 9665 Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 -𝑒𝐵) = (𝐴𝐵))

Theoremrexaddd 9666 The extended real addition operation when both arguments are real. Deduction version of rexadd 9664. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))

Theoremxnegcld 9667 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → -𝑒𝐴 ∈ ℝ*)

Theoremxrex 9668 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
* ∈ V

Theoremxaddnemnf 9669 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 9670 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 9671 Extended real version of negid 8032. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0)

Theoremxaddcl 9672 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 9673 The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = (𝐵 +𝑒 𝐴))

Theoremxaddid1 9674 Extended real version of addid1 7923. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴)

Theoremxaddid2 9675 Extended real version of addid2 7924. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴)

Theoremxaddid1d 9676 0 is a right identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → (𝐴 +𝑒 0) = 𝐴)

Theoremxnn0lenn0nn0 9677 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 9678 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 9679 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 9680 Extended real version of negdi 8042. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 𝐵) = (-𝑒𝐴 +𝑒 -𝑒𝐵))

Theoremxaddass 9681 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 9682, 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 9682 Associativity of extended real addition. See xaddass 9681 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 9683 Extended real version of pncan 7991. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → ((𝐴 +𝑒 𝐵) +𝑒 -𝑒𝐵) = 𝐴)

Theoremxnpcan 9684 Extended real version of npcan 7994. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)

Theoremxleadd1a 9685 Extended real version of leadd1 8215; 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 9686 Commuted form of xleadd1a 9685. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ 𝐴𝐵) → (𝐶 +𝑒 𝐴) ≤ (𝐶 +𝑒 𝐵))

Theoremxleadd1 9687 Weakened version of xleadd1a 9685 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 9688 Extended real version of ltadd1 8214. (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)))

Theoremxltadd2 9689 Extended real version of ltadd2 8204. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 +𝑒 𝐴) < (𝐶 +𝑒 𝐵)))

Theoremxaddge0 9690 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 9691 Extended real version of le2add 8229. (Contributed by Mario Carneiro, 23-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴𝐶𝐵𝐷) → (𝐴 +𝑒 𝐵) ≤ (𝐶 +𝑒 𝐷)))

Theoremxlt2add 9692 Extended real version of lt2add 8230. Note that ltleadd 8231, 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 9693 Extended real version of subge0 8260. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵𝐴))

Theoremxposdif 9694 Extended real version of posdif 8240. (Contributed by Mario Carneiro, 24-Aug-2015.) (Revised by Jim Kingdon, 17-Apr-2023.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +𝑒 -𝑒𝐴)))

Theoremxlesubadd 9695 Under certain conditions, the conclusion of lesubadd 8219 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)))

Theoremxaddcld 9696 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*)

Theoremxadd4d 9697 Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 7954. (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 9698 Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 9697. (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)))

Theoremxleaddadd 9699 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*) → (𝐴𝐵 ↔ (𝐴 +𝑒 𝐴) ≤ (𝐵 +𝑒 𝐵)))

4.5.3  Real number intervals

Syntaxcioo 9700 Extend class notation with the set of open intervals of extended reals.
class (,)

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