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Theorem List for Metamath Proof Explorer - 11901-12000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmax0sub 11901 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 11902 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 11903* 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 11904* 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 11905* 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 11906* 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 11907* Lemma for qextlt 11908 and qextle . (Contributed by Mario Carneiro, 3-Oct-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐴𝑥 < 𝐵) ∧ ¬ (𝑥𝐴𝑥𝐵))))

Theoremqextlt 11908* 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 11909* 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 11910* 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 11911* 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 11912 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 11913 A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)
-𝑒𝐴 ∈ V

Theoremxnegpnf 11914 Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)
-𝑒+∞ = -∞

Theoremxnegmnf 11915 Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)
-𝑒-∞ = +∞

Theoremrexneg 11916 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 11917 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)
-𝑒0 = 0

Theoremxnegcl 11918 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*)

Theoremxnegneg 11919 Extended real version of negneg 10210. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴)

Theoremxneg11 11920 Extended real version of neg11 10211. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵𝐴 = 𝐵))

Theoremxltnegi 11921 Forward direction of xltneg 11922. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴)

Theoremxltneg 11922 Extended real version of ltneg 10407. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴))

Theoremxleneg 11923 Extended real version of leneg 10410. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴))

Theoremxlt0neg1 11924 Extended real version of lt0neg1 10413. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴))

Theoremxlt0neg2 11925 Extended real version of lt0neg2 10414. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0))

Theoremxle0neg1 11926 Extended real version of le0neg1 10415. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴))

Theoremxle0neg2 11927 Extended real version of le0neg2 10416. (Contributed by Mario Carneiro, 9-Sep-2015.)
(𝐴 ∈ ℝ* → (0 ≤ 𝐴 ↔ -𝑒𝐴 ≤ 0))

Theoremxaddval 11928 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 11929 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
+𝑒 :(ℝ* × ℝ*)⟶ℝ*

Theoremxmulval 11930 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 11931 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)

Theoremxaddpnf2 11932 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞)

Theoremxaddmnf1 11933 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)

Theoremxaddmnf2 11934 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞)

Theorempnfaddmnf 11935 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 11936 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 11937 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 11938 Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 -𝑒𝐵) = (𝐴𝐵))

Theoremrexaddd 11939 The extended real addition operation when both arguments are real. Deduction version of rexadd 11937. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))

Theoremxnn0xaddcl 11940 The extended nonnegative integers are closed under extended addition. (Contributed by AV, 10-Dec-2020.)
((𝐴 ∈ ℕ0*𝐵 ∈ ℕ0*) → (𝐴 +𝑒 𝐵) ∈ ℕ0*)

Theoremxaddnemnf 11941 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 11942 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 11943 Extended real version of negid 10207. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0)

Theoremxaddcl 11944 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 11945 The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = (𝐵 +𝑒 𝐴))

Theoremxaddid1 11946 Extended real version of addid1 10095. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴)

Theoremxaddid2 11947 Extended real version of addid2 10098. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴)

Theoremxaddid1d 11948 0 is a right identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → (𝐴 +𝑒 0) = 𝐴)

Theoremxnn0xadd0 11949 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 11950 Extended real version of xnegdi 11950. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 𝐵) = (-𝑒𝐴 +𝑒 -𝑒𝐵))

Theoremxaddass 11951 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 11952, 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 11952 Associativity of extended real addition. See xaddass 11951 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 11953 Extended real version of pncan 10166. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → ((𝐴 +𝑒 𝐵) +𝑒 -𝑒𝐵) = 𝐴)

Theoremxnpcan 11954 Extended real version of npcan 10169. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)

Theoremxleadd1a 11955 Extended real version of leadd1 10375; 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 11956 Commuted form of xleadd1a 11955. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ 𝐴𝐵) → (𝐶 +𝑒 𝐴) ≤ (𝐶 +𝑒 𝐵))

Theoremxleadd1 11957 Weakened version of xleadd1a 11955 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 11958 Extended real version of ltadd1 10374. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 +𝑒 𝐶) < (𝐵 +𝑒 𝐶)))

Theoremxltadd2 11959 Extended real version of ltadd2 10020. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 +𝑒 𝐴) < (𝐶 +𝑒 𝐵)))

Theoremxaddge0 11960 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 11961 Extended real version of le2add 10389. (Contributed by Mario Carneiro, 23-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴𝐶𝐵𝐷) → (𝐴 +𝑒 𝐵) ≤ (𝐶 +𝑒 𝐷)))

Theoremxlt2add 11962 Extended real version of lt2add 10392. Note that ltleadd 10390, 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 11963 Extended real version of subge0 10420. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵𝐴))

Theoremxposdif 11964 Extended real version of posdif 10400. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +𝑒 -𝑒𝐴)))

Theoremxlesubadd 11965 Under certain conditions, the conclusion of lesubadd 10379 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)))

Theoremxmullem 11966 Lemma for rexmul 11973. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) ∧ ¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → 𝐴 ∈ ℝ)

Theoremxmullem2 11967 Lemma for xmulneg1 11971. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))

Theoremxmulcom 11968 Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) = (𝐵 ·e 𝐴))

Theoremxmul01 11969 Extended real version of mul01 10094. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0)

Theoremxmul02 11970 Extended real version of mul02 10093. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (0 ·e 𝐴) = 0)

Theoremxmulneg1 11971 Extended real version of mulneg1 10345. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = -𝑒(𝐴 ·e 𝐵))

Theoremxmulneg2 11972 Extended real version of mulneg2 10346. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e -𝑒𝐵) = -𝑒(𝐴 ·e 𝐵))

Theoremrexmul 11973 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))

Theoremxmulf 11974 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
·e :(ℝ* × ℝ*)⟶ℝ*

Theoremxmulcl 11975 Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) ∈ ℝ*)

Theoremxmulpnf1 11976 Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e +∞) = +∞)

Theoremxmulpnf2 11977 Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (+∞ ·e 𝐴) = +∞)

Theoremxmulmnf1 11978 Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e -∞) = -∞)

Theoremxmulmnf2 11979 Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (-∞ ·e 𝐴) = -∞)

Theoremxmulpnf1n 11980 Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐴 < 0) → (𝐴 ·e +∞) = -∞)

Theoremxmulid1 11981 Extended real version of mulid1 9916. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (𝐴 ·e 1) = 𝐴)

Theoremxmulid2 11982 Extended real version of mulid2 9917. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (1 ·e 𝐴) = 𝐴)

Theoremxmulm1 11983 Extended real version of mulm1 10350. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (-1 ·e 𝐴) = -𝑒𝐴)

Theoremxmulasslem2 11984 Lemma for xmulass 11989. (Contributed by Mario Carneiro, 20-Aug-2015.)
((0 < 𝐴𝐴 = -∞) → 𝜑)

Theoremxmulgt0 11985 Extended real version of mulgt0 9994. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 < 𝐵)) → 0 < (𝐴 ·e 𝐵))

Theoremxmulge0 11986 Extended real version of mulge0 10425. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 ·e 𝐵))

Theoremxmulasslem 11987* Lemma for xmulass 11989. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝑥 = 𝐷 → (𝜓𝑋 = 𝑌))    &   (𝑥 = -𝑒𝐷 → (𝜓𝐸 = 𝐹))    &   (𝜑𝑋 ∈ ℝ*)    &   (𝜑𝑌 ∈ ℝ*)    &   (𝜑𝐷 ∈ ℝ*)    &   ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝜓)    &   (𝜑 → (𝑥 = 0 → 𝜓))    &   (𝜑𝐸 = -𝑒𝑋)    &   (𝜑𝐹 = -𝑒𝑌)       (𝜑𝑋 = 𝑌)

Theoremxmulasslem3 11988 Lemma for xmulass 11989. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ* ∧ 0 < 𝐶)) → ((𝐴 ·e 𝐵) ·e 𝐶) = (𝐴 ·e (𝐵 ·e 𝐶)))

Theoremxmulass 11989 Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 11951 which has to avoid the "undefined" combinations +∞ +𝑒 -∞ and -∞ +𝑒 +∞. The equivalent "undefined" expression here would be 0 ·e +∞, but since this is defined to equal 0 any zeroes in the expression make the whole thing evaluate to zero (on both sides), thus establishing the identity in this case. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 ·e 𝐵) ·e 𝐶) = (𝐴 ·e (𝐵 ·e 𝐶)))

Theoremxlemul1a 11990 Extended real version of lemul1a 10756. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶)) ∧ 𝐴𝐵) → (𝐴 ·e 𝐶) ≤ (𝐵 ·e 𝐶))

Theoremxlemul2a 11991 Extended real version of lemul2a 10757. (Contributed by Mario Carneiro, 8-Sep-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶)) ∧ 𝐴𝐵) → (𝐶 ·e 𝐴) ≤ (𝐶 ·e 𝐵))

Theoremxlemul1 11992 Extended real version of lemul1 10754. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ+) → (𝐴𝐵 ↔ (𝐴 ·e 𝐶) ≤ (𝐵 ·e 𝐶)))

Theoremxlemul2 11993 Extended real version of lemul2 10755. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ+) → (𝐴𝐵 ↔ (𝐶 ·e 𝐴) ≤ (𝐶 ·e 𝐵)))

Theoremxltmul1 11994 Extended real version of ltmul1 10752. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐴 ·e 𝐶) < (𝐵 ·e 𝐶)))

Theoremxltmul2 11995 Extended real version of ltmul2 10753. (Contributed by Mario Carneiro, 8-Sep-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (𝐶 ·e 𝐴) < (𝐶 ·e 𝐵)))

(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ 0 < 𝐴) → (𝐴 ·e (𝐵 +𝑒 𝐶)) = ((𝐴 ·e 𝐵) +𝑒 (𝐴 ·e 𝐶)))

Theoremxadddi 11997 Distributive property for extended real addition and multiplication. Like xaddass 11951, this has an unusual domain of correctness due to counterexamples like (+∞ · (2 − 1)) = -∞ ≠ ((+∞ · 2) − (+∞ · 1)) = (+∞ − +∞) = 0. In this theorem we show that if the multiplier is real then everything works as expected. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐴 ·e (𝐵 +𝑒 𝐶)) = ((𝐴 ·e 𝐵) +𝑒 (𝐴 ·e 𝐶)))

Theoremxadddir 11998 Commuted version of xadddi 11997. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ) → ((𝐴 +𝑒 𝐵) ·e 𝐶) = ((𝐴 ·e 𝐶) +𝑒 (𝐵 ·e 𝐶)))

Theoremxadddi2 11999 The assumption that the multiplier be real in xadddi 11997 can be relaxed if the addends have the same sign. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ* ∧ 0 ≤ 𝐶)) → (𝐴 ·e (𝐵 +𝑒 𝐶)) = ((𝐴 ·e 𝐵) +𝑒 (𝐴 ·e 𝐶)))

Theoremxadddi2r 12000 Commuted version of xadddi2 11999. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ*) → ((𝐴 +𝑒 𝐵) ·e 𝐶) = ((𝐴 ·e 𝐶) +𝑒 (𝐵 ·e 𝐶)))

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