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Theorem List for Intuitionistic Logic Explorer - 9801-9900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelxr 9801 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
 
Theoremxrnemnf 9802 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
 
Theoremxrnepnf 9803 An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
 
Theoremxrltnr 9804 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
 
Theoremltpnf 9805 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ → 𝐴 < +∞)
 
Theoremltpnfd 9806 Any (finite) real is less than plus infinity. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 < +∞)
 
Theorem0ltpnf 9807 Zero is less than plus infinity (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 < +∞
 
Theoremmnflt 9808 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ → -∞ < 𝐴)
 
Theoremmnflt0 9809 Minus infinity is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-∞ < 0
 
Theoremmnfltpnf 9810 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
-∞ < +∞
 
Theoremmnfltxr 9811 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 9812 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
(𝐴 ∈ ℝ* → ¬ +∞ < 𝐴)
 
Theoremnltmnf 9813 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
(𝐴 ∈ ℝ* → ¬ 𝐴 < -∞)
 
Theorempnfge 9814 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
(𝐴 ∈ ℝ*𝐴 ≤ +∞)
 
Theorem0lepnf 9815 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≤ +∞
 
Theoremnn0pnfge0 9816 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 9817 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
(𝐴 ∈ ℝ* → -∞ ≤ 𝐴)
 
Theoremxrltnsym 9818 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
 
Theoremxrltnsym2 9819 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ¬ (𝐴 < 𝐵𝐵 < 𝐴))
 
Theoremxrlttr 9820 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 9821 'Less than' is a weakly linear ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
< Or ℝ*
 
Theoremxrlttri3 9822 Extended real version of lttri3 8062. (Contributed by NM, 9-Feb-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
 
Theoremxrltle 9823 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵𝐴𝐵))
 
Theoremxrltled 9824 'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle 9823. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴𝐵)
 
Theoremxrleid 9825 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
(𝐴 ∈ ℝ*𝐴𝐴)
 
Theoremxrleidd 9826 'Less than or equal to' is reflexive for extended reals. Deduction form of xrleid 9825. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)       (𝜑𝐴𝐴)
 
Theoremxnn0dcle 9827 Decidability of for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0*𝐵 ∈ ℕ0*) → DECID 𝐴𝐵)
 
Theoremxnn0letri 9828 Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝐴 ∈ ℕ0*𝐵 ∈ ℕ0*) → (𝐴𝐵𝐵𝐴))
 
Theoremxrletri3 9829 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴)))
 
Theoremxrletrid 9830 Trichotomy law for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐴)       (𝜑𝐴 = 𝐵)
 
Theoremxrlelttr 9831 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
 
Theoremxrltletr 9832 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 < 𝐵𝐵𝐶) → 𝐴 < 𝐶))
 
Theoremxrletr 9833 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremxrlttrd 9834 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 9835 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 9836 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 9837 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremxrltne 9838 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → 𝐵𝐴)
 
Theoremnltpnft 9839 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
(𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
 
Theoremnpnflt 9840 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 9841 An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ* → (+∞ ≤ 𝐴𝐴 = +∞))
 
Theoremngtmnft 9842 An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)
(𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))
 
Theoremnmnfgt 9843 An extended real is greater than minus infinite iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
(𝐴 ∈ ℝ* → (-∞ < 𝐴𝐴 ≠ -∞))
 
Theoremxrrebnd 9844 An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)
(𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴𝐴 < +∞)))
 
Theoremxrre 9845 A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ) ∧ (-∞ < 𝐴𝐴𝐵)) → 𝐴 ∈ ℝ)
 
Theoremxrre2 9846 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 9847 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ) ∧ (𝐵𝐴𝐴 < +∞)) → 𝐴 ∈ ℝ)
 
Theoremge0gtmnf 9848 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)
 
Theoremge0nemnf 9849 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞)
 
Theoremxrrege0 9850 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 9851* 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 9852 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 9853 Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
-𝑒+∞ = -∞
 
Theoremxnegmnf 9854 Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
-𝑒-∞ = +∞
 
Theoremrexneg 9855 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 9856 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
-𝑒0 = 0
 
Theoremxnegcl 9857 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*)
 
Theoremxnegneg 9858 Extended real version of negneg 8232. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴)
 
Theoremxneg11 9859 Extended real version of neg11 8233. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵𝐴 = 𝐵))
 
Theoremxltnegi 9860 Forward direction of xltneg 9861. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴)
 
Theoremxltneg 9861 Extended real version of ltneg 8444. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴))
 
Theoremxleneg 9862 Extended real version of leneg 8447. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴))
 
Theoremxlt0neg1 9863 Extended real version of lt0neg1 8450. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴))
 
Theoremxlt0neg2 9864 Extended real version of lt0neg2 8451. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0))
 
Theoremxle0neg1 9865 Extended real version of le0neg1 8452. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴))
 
Theoremxle0neg2 9866 Extended real version of le0neg2 8453. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (0 ≤ 𝐴 ↔ -𝑒𝐴 ≤ 0))
 
Theoremxrpnfdc 9867 An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
(𝐴 ∈ ℝ*DECID 𝐴 = +∞)
 
Theoremxrmnfdc 9868 An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
(𝐴 ∈ ℝ*DECID 𝐴 = -∞)
 
Theoremxaddf 9869 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
+𝑒 :(ℝ* × ℝ*)⟶ℝ*
 
Theoremxaddval 9870 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 9871 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)
 
Theoremxaddpnf2 9872 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞)
 
Theoremxaddmnf1 9873 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)
 
Theoremxaddmnf2 9874 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞)
 
Theorempnfaddmnf 9875 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 9876 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 9877 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 9878 Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 -𝑒𝐵) = (𝐴𝐵))
 
Theoremrexaddd 9879 The extended real addition operation when both arguments are real. Deduction version of rexadd 9877. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))
 
Theoremxnegcld 9880 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → -𝑒𝐴 ∈ ℝ*)
 
Theoremxrex 9881 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
* ∈ V
 
Theoremxaddnemnf 9882 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 9883 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 9884 Extended real version of negid 8229. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0)
 
Theoremxaddcl 9885 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 9886 The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = (𝐵 +𝑒 𝐴))
 
Theoremxaddid1 9887 Extended real version of addrid 8120. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴)
 
Theoremxaddid2 9888 Extended real version of addlid 8121. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴)
 
Theoremxaddid1d 9889 0 is a right identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → (𝐴 +𝑒 0) = 𝐴)
 
Theoremxnn0lenn0nn0 9890 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 9891 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 9892 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 9893 Extended real version of negdi 8239. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 𝐵) = (-𝑒𝐴 +𝑒 -𝑒𝐵))
 
Theoremxaddass 9894 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 9895, 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 9895 Associativity of extended real addition. See xaddass 9894 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 9896 Extended real version of pncan 8188. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → ((𝐴 +𝑒 𝐵) +𝑒 -𝑒𝐵) = 𝐴)
 
Theoremxnpcan 9897 Extended real version of npcan 8191. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)
 
Theoremxleadd1a 9898 Extended real version of leadd1 8412; 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 9899 Commuted form of xleadd1a 9898. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ 𝐴𝐵) → (𝐶 +𝑒 𝐴) ≤ (𝐶 +𝑒 𝐵))
 
Theoremxleadd1 9900 Weakened version of xleadd1a 9898 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)))
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