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Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem1red 7801 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℝ)

Theorem1cnd 7802 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(𝜑 → 1 ∈ ℂ)

Theoremmulid1d 7803 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · 1) = 𝐴)

Theoremmulid2d 7804 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 7806 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 · 𝐵) ∈ ℂ)

Theoremmulcomd 7807 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 7809 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 7810 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 7811 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 7812 Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵))

Theoremjoinlmuladdmuld 7813 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 7814 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℂ)

Theoremreaddcld 7815 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 + 𝐵) ∈ ℝ)

Theoremremulcld 7816 Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 · 𝐵) ∈ ℝ)

4.2.2  Infinity and the extended real number system

Syntaxcpnf 7817 Plus infinity.
class +∞

Syntaxcmnf 7818 Minus infinity.
class -∞

Syntaxcxr 7819 The set of extended reals (includes plus and minus infinity).
class *

Syntaxclt 7820 'Less than' predicate (extended to include the extended reals).
class <

Syntaxcle 7821 Extend wff notation to include the 'less than or equal to' relation.
class

Definitiondf-pnf 7822 Define plus infinity. Note that the definition is arbitrary, requiring only that +∞ be a set not in and different from -∞ (df-mnf 7823). 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 7827 and mnfnre 7828, and we'll also be able to prove +∞ ≠ -∞.

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 7823 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 7828). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
-∞ = 𝒫 +∞

Definitiondf-xr 7824 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 7825* 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 7826 Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.)
≤ = ((ℝ* × ℝ*) ∖ < )

Theorempnfnre 7827 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
+∞ ∉ ℝ

Theoremmnfnre 7828 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
-∞ ∉ ℝ

Theoremressxr 7829 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
ℝ ⊆ ℝ*

Theoremrexpssxrxp 7830 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 7831 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
(𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)

Theorem0xr 7832 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
0 ∈ ℝ*

Theoremrenepnf 7833 No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(𝐴 ∈ ℝ → 𝐴 ≠ +∞)

Theoremrenemnf 7834 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(𝐴 ∈ ℝ → 𝐴 ≠ -∞)

Theoremrexrd 7835 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℝ*)

Theoremrenepnfd 7836 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≠ +∞)

Theoremrenemnfd 7837 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≠ -∞)

Theorempnfxr 7838 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 7839 Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
+∞ ∈ V

Theorempnfnemnf 7840 Plus and minus infinity are different elements of *. (Contributed by NM, 14-Oct-2005.)
+∞ ≠ -∞

Theoremmnfnepnf 7841 Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-∞ ≠ +∞

Theoremmnfxr 7842 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 7843 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
𝐴 ∈ ℝ       𝐴 ∈ ℝ*

Theoremrenfdisj 7844 The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(ℝ ∩ {+∞, -∞}) = ∅

Theoremltrelxr 7845 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
< ⊆ (ℝ* × ℝ*)

Theoremltrel 7846 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
Rel <

Theoremlerelxr 7847 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
≤ ⊆ (ℝ* × ℝ*)

Theoremlerel 7848 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
Rel ≤

Theoremxrlenlt 7849 '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))

Theoremltxrlt 7850 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))

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

Theoremaxltirr 7851 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7752 with ordering on the extended reals. New proofs should use ltnr 7861 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
(𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)

Theoremaxltwlin 7852 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7753 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶𝐶 < 𝐵)))

Theoremaxlttrn 7853 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7754 with ordering on the extended reals. New proofs should use lttr 7858 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 7854 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7756 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))

Theoremaxapti 7855 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7755 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)

Theoremaxmulgt0 7856 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7757 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))

Theoremaxsuploc 7857* An inhabited, bounded-above, located set of reals has a supremum. Axiom for real and complex numbers, derived from ZF set theory. (This restates ax-pre-suploc 7761 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.)
(((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

4.2.4  Ordering on reals

Theoremlttr 7858 Alias for axlttrn 7853, for naming consistency with lttri 7888. New proofs should generally use this instead of ax-pre-lttrn 7754. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))

Theoremmulgt0 7859 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 7860 'Less than or equal to' expressed in terms of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-May-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))

Theoremltnr 7861 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
(𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)

Theoremltso 7862 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
< Or ℝ

Theoremgtso 7863 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
< Or ℝ

Theoremlttri3 7864 Tightness of real apartness. (Contributed by NM, 5-May-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))

Theoremletri3 7865 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴)))

Theoremltleletr 7866 Transitive law, weaker form of (𝐴 < 𝐵𝐵𝐶) → 𝐴 < 𝐶. (Contributed by AV, 14-Oct-2018.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵𝐶) → 𝐴𝐶))

Theoremletr 7867 Transitive law. (Contributed by NM, 12-Nov-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))

Theoremleid 7868 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
(𝐴 ∈ ℝ → 𝐴𝐴)

Theoremltne 7869 'Less than' implies not equal. See also ltap 8415 which is the same but for apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)
((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵𝐴)

Theoremltnsym 7870 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))

Theoremltle 7871 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴𝐵))

Theoremlelttr 7872 Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))

Theoremltletr 7873 Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵𝐶) → 𝐴 < 𝐶))

Theoremltnsym2 7874 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ (𝐴 < 𝐵𝐵 < 𝐴))

Theoremeqle 7875 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴𝐵)

Theoremltnri 7876 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
𝐴 ∈ ℝ        ¬ 𝐴 < 𝐴

Theoremeqlei 7877 Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.)
𝐴 ∈ ℝ       (𝐴 = 𝐵𝐴𝐵)

Theoremeqlei2 7878 Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
𝐴 ∈ ℝ       (𝐵 = 𝐴𝐵𝐴)

Theoremgtneii 7879 'Less than' implies not equal. See also gtapii 8416 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)
𝐴 ∈ ℝ    &   𝐴 < 𝐵       𝐵𝐴

Theoremltneii 7880 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
𝐴 ∈ ℝ    &   𝐴 < 𝐵       𝐴𝐵

Theoremlttri3i 7881 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))

Theoremletri3i 7882 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))

Theoremltnsymi 7883 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)

Theoremlenlti 7884 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴)

Theoremltlei 7885 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵𝐴𝐵)

Theoremltleii 7886 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐴 < 𝐵       𝐴𝐵

Theoremltnei 7887 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵𝐵𝐴)

Theoremlttri 7888 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶)

Theoremlelttri 7889 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵𝐵 < 𝐶) → 𝐴 < 𝐶)

Theoremltletri 7890 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴 < 𝐵𝐵𝐶) → 𝐴 < 𝐶)

Theoremletri 7891 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremle2tri3i 7892 Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵𝐵𝐶𝐶𝐴) ↔ (𝐴 = 𝐵𝐵 = 𝐶𝐶 = 𝐴))

Theoremmulgt0i 7893 The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵))

Theoremmulgt0ii 7894 The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   0 < 𝐴    &   0 < 𝐵       0 < (𝐴 · 𝐵)

Theoremltnrd 7895 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → ¬ 𝐴 < 𝐴)

Theoremgtned 7896 'Less than' implies not equal. See also gtapd 8419 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵𝐴)

Theoremltned 7897 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴𝐵)

Theoremlttri3d 7898 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))

Theoremletri3d 7899 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴)))

Theoremlenltd 7900 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))

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