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Theorem List for Metamath Proof Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmulcom 9901 Alias for ax-mulcom 9879, for naming consistency with mulcomi 9925. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))

((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Theoremmulass 9903 Alias for ax-mulass 9881, for naming consistency with mulassi 9928. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Theoremadddi 9904 Alias for ax-distr 9882, for naming consistency with adddii 9929. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Theoremrecn 9905 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
(𝐴 ∈ ℝ → 𝐴 ∈ ℂ)

Theoremreex 9906 The real numbers form a set. See also reexALT 11702. (Contributed by Mario Carneiro, 17-Nov-2014.)
ℝ ∈ V

Theoremreelprrecn 9907 Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
ℝ ∈ {ℝ, ℂ}

Theoremcnelprrecn 9908 Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
ℂ ∈ {ℝ, ℂ}

Theoremelimne0 9909 Hypothesis for weak deduction theorem to eliminate 𝐴 ≠ 0. (Contributed by NM, 15-May-1999.)
if(𝐴 ≠ 0, 𝐴, 1) ≠ 0

Theoremadddir 9910 Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Theorem0cn 9911 0 is a complex number. See also 0cnALT 10149. (Contributed by NM, 19-Feb-2005.)
0 ∈ ℂ

Theorem0cnd 9912 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 0 ∈ ℂ)

Theoremc0ex 9913 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ∈ V

Theorem1ex 9914 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
1 ∈ V

Theoremcnre 9915* Alias for ax-cnre 9888, for naming consistency. (Contributed by NM, 3-Jan-2013.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))

Theoremmulid1 9916 1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)

Theoremmulid2 9917 Identity law for multiplication. Note: see mulid1 9916 for commuted version. (Contributed by NM, 8-Oct-1999.)
(𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)

Theorem1re 9918 1 is a real number. This used to be one of our postulates for complex numbers, but Eric Schmidt discovered that it could be derived from a weaker postulate, ax-1cn 9873, by exploiting properties of the imaginary unit i. (Contributed by Eric Schmidt, 11-Apr-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
1 ∈ ℝ

Theorem0re 9919 0 is a real number. See also 0reALT 10257. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
0 ∈ ℝ

Theorem0red 9920 0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 0 ∈ ℝ)

Theoremmulid1i 9921 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
𝐴 ∈ ℂ       (𝐴 · 1) = 𝐴

Theoremmulid2i 9922 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
𝐴 ∈ ℂ       (1 · 𝐴) = 𝐴

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + 𝐵) ∈ ℂ

Theoremmulcli 9924 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 · 𝐵) ∈ ℂ

Theoremmulcomi 9925 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 · 𝐵) = (𝐵 · 𝐴)

Theoremmulcomli 9926 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 · 𝐵) = 𝐶       (𝐵 · 𝐴) = 𝐶

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))

Theoremmulassi 9928 Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))

Theoremadddii 9929 Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))

Theoremadddiri 9930 Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))

Theoremrecni 9931 A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
𝐴 ∈ ℝ       𝐴 ∈ ℂ

Theoremreaddcli 9932 Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 + 𝐵) ∈ ℝ

Theoremremulcli 9933 Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 · 𝐵) ∈ ℝ

Theorem1red 9934 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℝ)

Theorem1cnd 9935 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℂ)

Theoremmulid1d 9936 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · 1) = 𝐴)

Theoremmulid2d 9937 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (1 · 𝐴) = 𝐴)

(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 + 𝐵) ∈ ℂ)

Theoremmulcld 9939 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 · 𝐵) ∈ ℂ)

Theoremmulcomd 9940 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴))

(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Theoremmulassd 9942 Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Theoremadddid 9943 Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Theoremadddird 9944 Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Theoremadddirp1d 9945 Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵))

Theoremjoinlmuladdmuld 9946 Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷)       (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷)

Theoremrecnd 9947 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℂ)

Theoremreaddcld 9948 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 + 𝐵) ∈ ℝ)

Theoremremulcld 9949 Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 · 𝐵) ∈ ℝ)

5.2.2  Infinity and the extended real number system

Syntaxcpnf 9950 Plus infinity.
class +∞

Syntaxcmnf 9951 Minus infinity.
class -∞

Syntaxcxr 9952 The set of extended reals (includes plus and minus infinity).
class *

Syntaxclt 9953 'Less than' predicate (extended to include the extended reals).
class <

Syntaxcle 9954 Extend wff notation to include the 'less than or equal to' relation.
class

Definitiondf-pnf 9955 Define plus infinity. Note that the definition is arbitrary, requiring only that +∞ be a set not in and different from -∞ (df-mnf 9956). We use 𝒫 to make it independent of the construction of , and Cantor's Theorem will show that it is different from any member of and therefore . See pnfnre 9960, mnfnre 9961, and pnfnemnf 9973.

A simpler possibility is to define +∞ as and -∞ as {ℂ}, but that approach requires the Axiom of Regularity to show that +∞ and -∞ are different from each other and from all members of . (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)

+∞ = 𝒫

Definitiondf-mnf 9956 Define minus infinity as the power set of plus infinity. Note that the definition is arbitrary, requiring only that -∞ be a set not in and different from +∞ (see mnfnre 9961 and pnfnemnf 9973). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
-∞ = 𝒫 +∞

Definitiondf-xr 9957 Define the set of extended reals that includes plus and minus infinity. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 13-Oct-2005.)
* = (ℝ ∪ {+∞, -∞})

Definitiondf-ltxr 9958* Define 'less than' on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. Note that in our postulates for complex numbers, < is primitive and not necessarily a relation on . (Contributed by NM, 13-Oct-2005.)
< = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))

Definitiondf-le 9959 Define 'less than or equal to' on the extended real subset of complex numbers. Theorem leloe 10003 relates it to 'less than' for reals. (Contributed by NM, 13-Oct-2005.)
≤ = ((ℝ* × ℝ*) ∖ < )

Theorempnfnre 9960 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
+∞ ∉ ℝ

Theoremmnfnre 9961 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
-∞ ∉ ℝ

Theoremressxr 9962 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
ℝ ⊆ ℝ*

Theoremrexpssxrxp 9963 The Cartesian product of standard reals are a subset of the Cartesian product of extended reals (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(ℝ × ℝ) ⊆ (ℝ* × ℝ*)

Theoremrexr 9964 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)

Theorem0xr 9965 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
0 ∈ ℝ*

Theoremrenepnf 9966 No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(𝐴 ∈ ℝ → 𝐴 ≠ +∞)

Theoremrenemnf 9967 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(𝐴 ∈ ℝ → 𝐴 ≠ -∞)

Theoremrexrd 9968 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℝ*)

Theoremrenepnfd 9969 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≠ +∞)

Theoremrenemnfd 9970 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≠ -∞)

Theorempnfxr 9971 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)
+∞ ∈ ℝ*

Theorempnfex 9972 Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
+∞ ∈ V

Theorempnfnemnf 9973 Plus and minus infinity are different elements of *. (Contributed by NM, 14-Oct-2005.)
+∞ ≠ -∞

Theoremmnfnepnf 9974 Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-∞ ≠ +∞

Theoremmnfxr 9975 Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
-∞ ∈ ℝ*

Theoremrexri 9976 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
𝐴 ∈ ℝ       𝐴 ∈ ℝ*

Theoremrenfdisj 9977 The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(ℝ ∩ {+∞, -∞}) = ∅

Theoremltrelxr 9978 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
< ⊆ (ℝ* × ℝ*)

Theoremltrel 9979 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
Rel <

Theoremlerelxr 9980 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
≤ ⊆ (ℝ* × ℝ*)

Theoremlerel 9981 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
Rel ≤

Theoremxrlenlt 9982 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))

Theoremxrlenltd 9983 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))

Theoremxrltnle 9984 'Less than' expressed in terms of 'less than or equal to', for extended reals. (Contributed by NM, 6-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))

Theoremxrnltled 9985 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑 → ¬ 𝐵 < 𝐴)       (𝜑𝐴𝐵)

Theoremssxr 9986 The three (non-exclusive) possibilities implied by a subset of extended reals. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))

Theoremltxrlt 9987 The standard less-than < and the extended real less-than < are identical when restricted to the non-extended reals . (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴 < 𝐵))

5.2.3  Restate the ordering postulates with extended real "less than"

Theoremaxlttri 9988 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-lttri 9889 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))

Theoremaxlttrn 9989 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. This restates ax-pre-lttrn 9890 with ordering on the extended reals. New proofs should use lttr 9993 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))

Theoremaxltadd 9990 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-ltadd 9891 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))

Theoremaxmulgt0 9991 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-mulgt0 9892 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))

Theoremaxsup 9992* A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-sup 9893 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

5.2.4  Ordering on reals

Theoremlttr 9993 Alias for axlttrn 9989, for naming consistency with lttri 10042. New proofs should generally use this instead of ax-pre-lttrn 9890. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))

Theoremmulgt0 9994 The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.)
(((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵))

Theoremlenlt 9995 'Less than or equal to' expressed in terms of 'less than'. (Contributed by NM, 13-May-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))

Theoremltnle 9996 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ 𝐵𝐴))

Theoremltso 9997 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
< Or ℝ

Theoremgtso 9998 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
< Or ℝ

Theoremlttri2 9999 Consequence of trichotomy. (Contributed by NM, 9-Oct-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))

Theoremlttri3 10000 Trichotomy law for 'less than'. (Contributed by NM, 5-May-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))

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