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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem0xr 7101 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
0 ∈ ℝ*
 
Theoremrenepnf 7102 No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(𝐴 ∈ ℝ → 𝐴 ≠ +∞)
 
Theoremrenemnf 7103 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(𝐴 ∈ ℝ → 𝐴 ≠ -∞)
 
Theoremrexrd 7104 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℝ*)
 
Theoremrenepnfd 7105 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≠ +∞)
 
Theoremrenemnfd 7106 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≠ -∞)
 
Theoremrexri 7107 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
𝐴 ∈ ℝ       𝐴 ∈ ℝ*
 
Theoremrenfdisj 7108 The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(ℝ ∩ {+∞, -∞}) = ∅
 
Theoremltrelxr 7109 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
< ⊆ (ℝ* × ℝ*)
 
Theoremltrel 7110 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)
Rel <
 
Theoremlerelxr 7111 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
≤ ⊆ (ℝ* × ℝ*)
 
Theoremlerel 7112 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
Rel ≤
 
Theoremxrlenlt 7113 '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 7114 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))
 
3.2.3  Restate the ordering postulates with extended real "less than"
 
Theoremaxltirr 7115 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7024 with ordering on the extended reals. New proofs should use ltnr 7124 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
(𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
 
Theoremaxltwlin 7116 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7025 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 7117 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7026 with ordering on the extended reals. New proofs should use lttr 7121 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 7118 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7028 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 7119 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7027 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)
 
Theoremaxmulgt0 7120 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7029 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)))
 
3.2.4  Ordering on reals
 
Theoremlttr 7121 Alias for axlttrn 7117, for naming consistency with lttri 7151. New proofs should generally use this instead of ax-pre-lttrn 7026. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
 
Theoremmulgt0 7122 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 7123 '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 7124 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
(𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
 
Theoremltso 7125 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
< Or ℝ
 
Theoremgtso 7126 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
< Or ℝ
 
Theoremlttri3 7127 Tightness of real apartness. (Contributed by NM, 5-May-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
 
Theoremletri3 7128 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴)))
 
Theoremltleletr 7129 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 7130 Transitive law. (Contributed by NM, 12-Nov-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremleid 7131 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
(𝐴 ∈ ℝ → 𝐴𝐴)
 
Theoremltne 7132 'Less than' implies not equal. See also ltap 7666 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 7133 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
 
Theoremltle 7134 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴𝐵))
 
Theoremlelttr 7135 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 7136 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 7137 '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 7138 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴𝐵)
 
Theoremltnri 7139 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
𝐴 ∈ ℝ        ¬ 𝐴 < 𝐴
 
Theoremeqlei 7140 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 7141 Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
𝐴 ∈ ℝ       (𝐵 = 𝐴𝐵𝐴)
 
Theoremgtneii 7142 'Less than' implies not equal. See also gtapii 7667 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)
𝐴 ∈ ℝ    &   𝐴 < 𝐵       𝐵𝐴
 
Theoremltneii 7143 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
𝐴 ∈ ℝ    &   𝐴 < 𝐵       𝐴𝐵
 
Theoremlttri3i 7144 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
 
Theoremletri3i 7145 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
 
Theoremltnsymi 7146 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)
 
Theoremlenlti 7147 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴)
 
Theoremltlei 7148 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵𝐴𝐵)
 
Theoremltleii 7149 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐴 < 𝐵       𝐴𝐵
 
Theoremltnei 7150 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵𝐵𝐴)
 
Theoremlttri 7151 '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 7152 '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 7153 '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 7154 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremle2tri3i 7155 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 7156 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 7157 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 7158 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → ¬ 𝐴 < 𝐴)
 
Theoremgtned 7159 'Less than' implies not equal. See also gtapd 7670 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵𝐴)
 
Theoremltned 7160 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴𝐵)
 
Theoremlttri3d 7161 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
 
Theoremletri3d 7162 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴)))
 
Theoremlenltd 7163 '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 7164 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴𝐵)
 
Theoremltnsymd 7165 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → ¬ 𝐵 < 𝐴)
 
Theoremnltled 7166 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ¬ 𝐵 < 𝐴)       (𝜑𝐴𝐵)
 
Theoremlensymd 7167 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → ¬ 𝐵 < 𝐴)
 
Theoremmulgt0d 7168 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 7169 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 7170 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 7171 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 7172 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 7173 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
 
Theoremmul32 7174 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))
 
Theoremmul31 7175 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))
 
Theoremmul4 7176 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷)))
 
Theoremmuladd11 7177 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = ((1 + 𝐴) + (𝐵 + (𝐴 · 𝐵))))
 
Theorem1p1times 7178 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴))
 
Theorempeano2cn 7179 A theorem for complex numbers analogous the second Peano postulate peano2 4343. (Contributed by NM, 17-Aug-2005.)
(𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)
 
Theorempeano2re 7180 A theorem for reals analogous the second Peano postulate peano2 4343. (Contributed by NM, 5-Jul-2005.)
(𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)
 
Theoremaddcom 7181 Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theoremaddid1 7182 0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
(𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
 
Theoremaddid2 7183 0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
 
Theoremreaddcan 7184 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) ↔ 𝐴 = 𝐵))
 
Theorem00id 7185 0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
(0 + 0) = 0
 
Theoremaddid1i 7186 0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
𝐴 ∈ ℂ       (𝐴 + 0) = 𝐴
 
Theoremaddid2i 7187 0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
𝐴 ∈ ℂ       (0 + 𝐴) = 𝐴
 
Theoremaddcomi 7188 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + 𝐵) = (𝐵 + 𝐴)
 
Theoremaddcomli 7189 Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 + 𝐵) = 𝐶       (𝐵 + 𝐴) = 𝐶
 
Theoremmul12i 7190 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))
 
Theoremmul32i 7191 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)
 
Theoremmul4i 7192 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ       ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))
 
Theoremaddid1d 7193 0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 + 0) = 𝐴)
 
Theoremaddid2d 7194 0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (0 + 𝐴) = 𝐴)
 
Theoremaddcomd 7195 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theoremmul12d 7196 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
 
Theoremmul32d 7197 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))
 
Theoremmul31d 7198 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))
 
Theoremmul4d 7199 Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷)))
 
Theoremmuladd11r 7200 A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 1) · (𝐵 + 1)) = (((𝐴 · 𝐵) + (𝐴 + 𝐵)) + 1))
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