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Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1cnd 6801 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
(φ → 1 ℂ)
 
Theoremmulid1d 6802 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A · 1) = A)
 
Theoremmulid2d 6803 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (1 · A) = A)
 
Theoremaddcld 6804 Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A + B) ℂ)
 
Theoremmulcld 6805 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A · B) ℂ)
 
Theoremmulcomd 6806 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A · B) = (B · A))
 
Theoremaddassd 6807 Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + B) + 𝐶) = (A + (B + 𝐶)))
 
Theoremmulassd 6808 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 6809 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 6810 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)))
 
Theoremrecnd 6811 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
(φA ℝ)       (φA ℂ)
 
Theoremreaddcld 6812 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (A + B) ℝ)
 
Theoremremulcld 6813 Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (A · B) ℝ)
 
3.2.2  Infinity and the extended real number system
 
Syntaxcpnf 6814 Plus infinity.
class +∞
 
Syntaxcmnf 6815 Minus infinity.
class -∞
 
Syntaxcxr 6816 The set of extended reals (includes plus and minus infinity).
class *
 
Syntaxclt 6817 'Less than' predicate (extended to include the extended reals).
class <
 
Syntaxcle 6818 Extend wff notation to include the 'less than or equal to' relation.
class
 
Definitiondf-pnf 6819 Define plus infinity. Note that the definition is arbitrary, requiring only that +∞ be a set not in and different from -∞ (df-mnf 6820). 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 6824 and mnfnre 6825, 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 6820 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 6825). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
-∞ = 𝒫 +∞
 
Definitiondf-xr 6821 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 6822* 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 y x < y)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
 
Definitiondf-le 6823 Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.)
≤ = ((ℝ* × ℝ*) ∖ < )
 
Theorempnfnre 6824 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
+∞ ∉ ℝ
 
Theoremmnfnre 6825 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
-∞ ∉ ℝ
 
Theoremressxr 6826 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
ℝ ⊆ ℝ*
 
Theoremrexpssxrxp 6827 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 6828 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
(A ℝ → A *)
 
Theorem0xr 6829 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
0 *
 
Theoremrenepnf 6830 No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(A ℝ → A ≠ +∞)
 
Theoremrenemnf 6831 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(A ℝ → A ≠ -∞)
 
Theoremrexrd 6832 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φA *)
 
Theoremrenepnfd 6833 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φA ≠ +∞)
 
Theoremrenemnfd 6834 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℝ)       (φA ≠ -∞)
 
Theoremrexri 6835 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
A        A *
 
Theoremrenfdisj 6836 The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(ℝ ∩ {+∞, -∞}) = ∅
 
Theoremltrelxr 6837 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
< ⊆ (ℝ* × ℝ*)
 
Theoremltrel 6838 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
Rel <
 
Theoremlerelxr 6839 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
≤ ⊆ (ℝ* × ℝ*)
 
Theoremlerel 6840 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
Rel ≤
 
Theoremxrlenlt 6841 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)
((A * B *) → (AB ↔ ¬ B < A))
 
Theoremltxrlt 6842 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 B ℝ) → (A < BA < B))
 
3.2.3  Restate the ordering postulates with extended real "less than"
 
Theoremaxltirr 6843 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 6755 with ordering on the extended reals. New proofs should use ltnr 6852 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
(A ℝ → ¬ A < A)
 
Theoremaxltwlin 6844 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 6756 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.)
((A B 𝐶 ℝ) → (A < B → (A < 𝐶 𝐶 < B)))
 
Theoremaxlttrn 6845 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 6757 with ordering on the extended reals. New proofs should use lttr 6849 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
((A B 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))
 
Theoremaxltadd 6846 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 6759 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
((A B 𝐶 ℝ) → (A < B → (𝐶 + A) < (𝐶 + B)))
 
Theoremaxapti 6847 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 6758 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
((A B ¬ (A < B B < A)) → A = B)
 
Theoremaxmulgt0 6848 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 6760 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
((A B ℝ) → ((0 < A 0 < B) → 0 < (A · B)))
 
3.2.4  Ordering on reals
 
Theoremlttr 6849 Alias for axlttrn 6845, for naming consistency with lttri 6879. New proofs should generally use this instead of ax-pre-lttrn 6757. (Contributed by NM, 10-Mar-2008.)
((A B 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))
 
Theoremmulgt0 6850 The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.)
(((A 0 < A) (B 0 < B)) → 0 < (A · B))
 
Theoremlenlt 6851 '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 B ℝ) → (AB ↔ ¬ B < A))
 
Theoremltnr 6852 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
(A ℝ → ¬ A < A)
 
Theoremltso 6853 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
< Or ℝ
 
Theoremgtso 6854 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
< Or ℝ
 
Theoremlttri3 6855 Tightness of real apartness. (Contributed by NM, 5-May-1999.)
((A B ℝ) → (A = B ↔ (¬ A < B ¬ B < A)))
 
Theoremletri3 6856 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
((A B ℝ) → (A = B ↔ (AB BA)))
 
Theoremltleletr 6857 Transitive law, weaker form of (A < B B𝐶) → A < 𝐶. (Contributed by AV, 14-Oct-2018.)
((A B 𝐶 ℝ) → ((A < B B𝐶) → A𝐶))
 
Theoremletr 6858 Transitive law. (Contributed by NM, 12-Nov-1999.)
((A B 𝐶 ℝ) → ((AB B𝐶) → A𝐶))
 
Theoremleid 6859 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
(A ℝ → AA)
 
Theoremltne 6860 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)
((A A < B) → BA)
 
Theoremltnsym 6861 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
((A B ℝ) → (A < B → ¬ B < A))
 
Theoremltle 6862 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)
((A B ℝ) → (A < BAB))
 
Theoremlelttr 6863 Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.)
((A B 𝐶 ℝ) → ((AB B < 𝐶) → A < 𝐶))
 
Theoremltletr 6864 Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.)
((A B 𝐶 ℝ) → ((A < B B𝐶) → A < 𝐶))
 
Theoremltnsym2 6865 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((A B ℝ) → ¬ (A < B B < A))
 
Theoremeqle 6866 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
((A A = B) → AB)
 
Theoremltnri 6867 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
A         ¬ A < A
 
Theoremeqlei 6868 Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.)
A        (A = BAB)
 
Theoremeqlei2 6869 Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
A        (B = ABA)
 
Theoremgtneii 6870 'Less than' implies not equal. (Contributed by Mario Carneiro, 30-Sep-2013.)
A     &   A < B       BA
 
Theoremltneii 6871 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
A     &   A < B       AB
 
Theoremlttri3i 6872 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
A     &   B        (A = B ↔ (¬ A < B ¬ B < A))
 
Theoremletri3i 6873 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
A     &   B        (A = B ↔ (AB BA))
 
Theoremltnsymi 6874 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
A     &   B        (A < B → ¬ B < A)
 
Theoremlenlti 6875 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)
A     &   B        (AB ↔ ¬ B < A)
 
Theoremltlei 6876 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)
A     &   B        (A < BAB)
 
Theoremltleii 6877 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)
A     &   B     &   A < B       AB
 
Theoremltnei 6878 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
A     &   B        (A < BBA)
 
Theoremlttri 6879 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶        ((A < B B < 𝐶) → A < 𝐶)
 
Theoremlelttri 6880 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶        ((AB B < 𝐶) → A < 𝐶)
 
Theoremltletri 6881 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶        ((A < B B𝐶) → A < 𝐶)
 
Theoremletri 6882 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶        ((AB B𝐶) → A𝐶)
 
Theoremle2tri3i 6883 Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)
A     &   B     &   𝐶        ((AB B𝐶 𝐶A) ↔ (A = B B = 𝐶 𝐶 = A))
 
Theoremmulgt0i 6884 The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
A     &   B        ((0 < A 0 < B) → 0 < (A · B))
 
Theoremmulgt0ii 6885 The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
A     &   B     &   0 < A    &   0 < B       0 < (A · B)
 
Theoremltnrd 6886 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → ¬ A < A)
 
Theoremgtned 6887 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φA < B)       (φBA)
 
Theoremltned 6888 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φA < B)       (φAB)
 
Theoremlttri3d 6889 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (A = B ↔ (¬ A < B ¬ B < A)))
 
Theoremletri3d 6890 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (A = B ↔ (AB BA)))
 
Theoremlenltd 6891 '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))
 
Theoremltled 6892 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φA < B)       (φAB)
 
Theoremltnsymd 6893 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φA < B)       (φ → ¬ B < A)
 
Theoremmulgt0d 6894 The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 0 < A)    &   (φ → 0 < B)       (φ → 0 < (A · B))
 
Theoremletrd 6895 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φAB)    &   (φB𝐶)       (φA𝐶)
 
Theoremlelttrd 6896 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φAB)    &   (φB < 𝐶)       (φA < 𝐶)
 
Theoremlttrd 6897 Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < B)    &   (φB < 𝐶)       (φA < 𝐶)
 
Theorem0lt1 6898 0 is less than 1. Theorem I.21 of [Apostol] p. 20. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 17-Jan-1997.)
0 < 1
 
3.2.5  Initial properties of the complex numbers
 
Theoremmul12 6899 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
((A B 𝐶 ℂ) → (A · (B · 𝐶)) = (B · (A · 𝐶)))
 
Theoremmul32 6900 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
((A B 𝐶 ℂ) → ((A · B) · 𝐶) = ((A · 𝐶) · B))
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