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Theorem List for Metamath Proof Explorer - 12001-12100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnltmnf 12001 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
(𝐴 ∈ ℝ* → ¬ 𝐴 < -∞)

Theorempnfge 12002 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
(𝐴 ∈ ℝ*𝐴 ≤ +∞)

Theoremxnn0n0n1ge2b 12003 An extended nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by AV, 5-Apr-2021.)
(𝑁 ∈ ℕ0* → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁))

Theorem0lepnf 12004 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≤ +∞

Theoremxnn0ge0 12005 An extended nonnegative integer is greater than or equal to 0. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 10-Dec-2020.)
(𝑁 ∈ ℕ0* → 0 ≤ 𝑁)

Theoremnn0pnfge0OLD 12006 Obsolete version of xnn0ge0 12005 as of 24-Oct-2021. 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.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑁 ∈ ℕ0𝑁 = +∞) → 0 ≤ 𝑁)

Theoremmnfle 12007 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
(𝐴 ∈ ℝ* → -∞ ≤ 𝐴)

Theoremxrltnsym 12008 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))

Theoremxrltnsym2 12009 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ¬ (𝐴 < 𝐵𝐵 < 𝐴))

Theoremxrlttri 12010 Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 10048 or axlttri 10147. (Contributed by NM, 14-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))

Theoremxrlttr 12011 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 12012 'Less than' is a strict ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)
< Or ℝ*

Theoremxrlttri2 12013 Trichotomy law for 'less than' for extended reals. (Contributed by NM, 10-Dec-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))

Theoremxrlttri3 12014 Trichotomy law for 'less than' for extended reals. (Contributed by NM, 9-Feb-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))

Theoremxrleloe 12015 'Less than or equal' expressed in terms of 'less than' or 'equals', for extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ (𝐴 < 𝐵𝐴 = 𝐵)))

Theoremxrleltne 12016 'Less than or equal to' implies 'less than' is not 'equals', for extended reals. (Contributed by NM, 9-Feb-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → (𝐴 < 𝐵𝐵𝐴))

Theoremxrltlen 12017 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ (𝐴𝐵𝐵𝐴)))

Theoremdfle2 12018 Alternative definition of 'less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 6-Nov-2015.)
≤ = ( < ∪ ( I ↾ ℝ*))

Theoremdflt2 12019 Alternative definition of 'less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)
< = ( ≤ ∖ I )

Theoremxrltle 12020 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵𝐴𝐵))

Theoremxrleid 12021 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)
(𝐴 ∈ ℝ*𝐴𝐴)

Theoremxrletri 12022 Trichotomy law for extended reals. (Contributed by NM, 7-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵𝐵𝐴))

Theoremxrletri3 12023 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴)))

Theoremxrletrid 12024 Trichotomy law for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐴)       (𝜑𝐴 = 𝐵)

Theoremxrlelttr 12025 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))

Theoremxrltletr 12026 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 < 𝐵𝐵𝐶) → 𝐴 < 𝐶))

Theoremxrletr 12027 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))

Theoremxrlttrd 12028 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 12029 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 12030 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 12031 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremxrltne 12032 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → 𝐵𝐴)

Theoremnltpnft 12033 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
(𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))

Theoremxgepnf 12034 An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ* → (+∞ ≤ 𝐴𝐴 = +∞))

Theoremngtmnft 12035 An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)
(𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))

Theoremxlemnf 12036 An extended real which is less than minus infinity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
(𝐴 ∈ ℝ* → (𝐴 ≤ -∞ ↔ 𝐴 = -∞))

Theoremxrrebnd 12037 An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)
(𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴𝐴 < +∞)))

Theoremxrre 12038 A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ) ∧ (-∞ < 𝐴𝐴𝐵)) → 𝐴 ∈ ℝ)

Theoremxrre2 12039 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 12040 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ) ∧ (𝐵𝐴𝐴 < +∞)) → 𝐴 ∈ ℝ)

Theoremge0gtmnf 12041 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)

Theoremge0nemnf 12042 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞)

Theoremxrrege0 12043 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)

Theoremxrmax1 12044 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))

Theoremxrmax2 12045 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐵 ≤ if(𝐴𝐵, 𝐵, 𝐴))

Theoremxrmin1 12046 The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → if(𝐴𝐵, 𝐴, 𝐵) ≤ 𝐴)

Theoremxrmin2 12047 The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → if(𝐴𝐵, 𝐴, 𝐵) ≤ 𝐵)

Theoremxrmaxeq 12048 The maximum of two extended reals is equal to the first if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐵𝐴) → if(𝐴𝐵, 𝐵, 𝐴) = 𝐴)

Theoremxrmineq 12049 The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐵𝐴) → if(𝐴𝐵, 𝐴, 𝐵) = 𝐵)

Theoremxrmaxlt 12050 Two ways of saying the maximum of two extended reals is less than a third. (Contributed by NM, 7-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (if(𝐴𝐵, 𝐵, 𝐴) < 𝐶 ↔ (𝐴 < 𝐶𝐵 < 𝐶)))

Theoremxrltmin 12051 Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐴 < if(𝐵𝐶, 𝐵, 𝐶) ↔ (𝐴 < 𝐵𝐴 < 𝐶)))

Theoremxrmaxle 12052 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (if(𝐴𝐵, 𝐵, 𝐴) ≤ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))

Theoremxrlemin 12053 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(𝐵𝐶, 𝐵, 𝐶) ↔ (𝐴𝐵𝐴𝐶)))

Theoremmax1 12054 A number is less than or equal to the maximum of it and another. See also max1ALT 12055. (Contributed by NM, 3-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))

Theoremmax1ALT 12055 A number is less than or equal to the maximum of it and another. This version of max1 12054 omits the 𝐵 ∈ ℝ antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 12054 is preferred. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 3-Apr-2005.)
(𝐴 ∈ ℝ → 𝐴 ≤ if(𝐴𝐵, 𝐵, 𝐴))

Theoremmax2 12056 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 12057 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 12058 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 12059 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 12060 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 12061 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 12062 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 12063 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 12064 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 12065 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 12066 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 12067* 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 12068* 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 12069* 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 12070* 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 12071* Lemma for qextlt 12072 and qextle . (Contributed by Mario Carneiro, 3-Oct-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐴𝑥 < 𝐵) ∧ ¬ (𝑥𝐴𝑥𝐵))))

Theoremqextlt 12072* 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 12073* 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 12074* 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 12075* 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 12076 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 12077 A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)
-𝑒𝐴 ∈ V

Theoremxnegpnf 12078 Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
-𝑒+∞ = -∞

Theoremxnegmnf 12079 Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
-𝑒-∞ = +∞

Theoremrexneg 12080 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 12081 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
-𝑒0 = 0

Theoremxnegcl 12082 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*)

Theoremxnegneg 12083 Extended real version of negneg 10369. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴)

Theoremxneg11 12084 Extended real version of neg11 10370. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵𝐴 = 𝐵))

Theoremxltnegi 12085 Forward direction of xltneg 12086. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴)

Theoremxltneg 12086 Extended real version of ltneg 10566. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴))

Theoremxleneg 12087 Extended real version of leneg 10569. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴))

Theoremxlt0neg1 12088 Extended real version of lt0neg1 10572. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴))

Theoremxlt0neg2 12089 Extended real version of lt0neg2 10573. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0))

Theoremxle0neg1 12090 Extended real version of le0neg1 10574. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴))

Theoremxle0neg2 12091 Extended real version of le0neg2 10575. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (0 ≤ 𝐴 ↔ -𝑒𝐴 ≤ 0))

Theoremxaddval 12092 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 12093 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
+𝑒 :(ℝ* × ℝ*)⟶ℝ*

Theoremxmulval 12094 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 12095 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)

Theoremxaddpnf2 12096 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞)

Theoremxaddmnf1 12097 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)

Theoremxaddmnf2 12098 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞)

Theorempnfaddmnf 12099 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 12100 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

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