HomeHome Intuitionistic Logic Explorer
Theorem List (p. 78 of 129)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxrlenlt 7701 '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 7702 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 7703 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7607 with ordering on the extended reals. New proofs should use ltnr 7712 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
(𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
 
Theoremaxltwlin 7704 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7608 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 7705 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7609 with ordering on the extended reals. New proofs should use lttr 7709 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 7706 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7611 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 7707 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7610 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)
 
Theoremaxmulgt0 7708 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7612 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 7709 Alias for axlttrn 7705, for naming consistency with lttri 7739. New proofs should generally use this instead of ax-pre-lttrn 7609. (Contributed by NM, 10-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
 
Theoremmulgt0 7710 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 7711 '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 7712 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
(𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
 
Theoremltso 7713 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
< Or ℝ
 
Theoremgtso 7714 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
< Or ℝ
 
Theoremlttri3 7715 Tightness of real apartness. (Contributed by NM, 5-May-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
 
Theoremletri3 7716 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴)))
 
Theoremltleletr 7717 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 7718 Transitive law. (Contributed by NM, 12-Nov-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremleid 7719 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
(𝐴 ∈ ℝ → 𝐴𝐴)
 
Theoremltne 7720 'Less than' implies not equal. See also ltap 8260 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 7721 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
 
Theoremltle 7722 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵𝐴𝐵))
 
Theoremlelttr 7723 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 7724 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 7725 '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 7726 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴𝐵)
 
Theoremltnri 7727 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
𝐴 ∈ ℝ        ¬ 𝐴 < 𝐴
 
Theoremeqlei 7728 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 7729 Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
𝐴 ∈ ℝ       (𝐵 = 𝐴𝐵𝐴)
 
Theoremgtneii 7730 'Less than' implies not equal. See also gtapii 8261 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)
𝐴 ∈ ℝ    &   𝐴 < 𝐵       𝐵𝐴
 
Theoremltneii 7731 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
𝐴 ∈ ℝ    &   𝐴 < 𝐵       𝐴𝐵
 
Theoremlttri3i 7732 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
 
Theoremletri3i 7733 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
 
Theoremltnsymi 7734 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)
 
Theoremlenlti 7735 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴)
 
Theoremltlei 7736 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵𝐴𝐵)
 
Theoremltleii 7737 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐴 < 𝐵       𝐴𝐵
 
Theoremltnei 7738 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵𝐵𝐴)
 
Theoremlttri 7739 '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 7740 '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 7741 '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 7742 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremle2tri3i 7743 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 7744 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 7745 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 7746 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → ¬ 𝐴 < 𝐴)
 
Theoremgtned 7747 'Less than' implies not equal. See also gtapd 8264 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵𝐴)
 
Theoremltned 7748 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴𝐵)
 
Theoremlttri3d 7749 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
 
Theoremletri3d 7750 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴)))
 
Theoremlenltd 7751 '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 7752 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴𝐵)
 
Theoremltnsymd 7753 '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 7754 '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 7755 '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 7756 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 7757 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 7758 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 7759 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 7760 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
 
Theoremltntri 7761 Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, 𝐴 < 𝐵𝐴 = 𝐵𝐵 < 𝐴. Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴))
 
3.2.5  Initial properties of the complex numbers
 
Theoremmul12 7762 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
 
Theoremmul32 7763 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))
 
Theoremmul31 7764 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))
 
Theoremmul4 7765 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 7766 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 7767 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 7768 A theorem for complex numbers analogous the second Peano postulate peano2 4447. (Contributed by NM, 17-Aug-2005.)
(𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)
 
Theorempeano2re 7769 A theorem for reals analogous the second Peano postulate peano2 4447. (Contributed by NM, 5-Jul-2005.)
(𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)
 
Theoremaddcom 7770 Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theoremaddid1 7771 0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
(𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
 
Theoremaddid2 7772 0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
 
Theoremreaddcan 7773 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 7774 0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
(0 + 0) = 0
 
Theoremaddid1i 7775 0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
𝐴 ∈ ℂ       (𝐴 + 0) = 𝐴
 
Theoremaddid2i 7776 0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
𝐴 ∈ ℂ       (0 + 𝐴) = 𝐴
 
Theoremaddcomi 7777 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + 𝐵) = (𝐵 + 𝐴)
 
Theoremaddcomli 7778 Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 + 𝐵) = 𝐶       (𝐵 + 𝐴) = 𝐶
 
Theoremmul12i 7779 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 7780 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 7781 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 7782 0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 + 0) = 𝐴)
 
Theoremaddid2d 7783 0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (0 + 𝐴) = 𝐴)
 
Theoremaddcomd 7784 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 7785 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 7786 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 7787 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))
 
Theoremmul4d 7788 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 7789 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))
 
Theoremcomraddd 7790 Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 = (𝐵 + 𝐶))       (𝜑𝐴 = (𝐶 + 𝐵))
 
3.3  Real and complex numbers - basic operations
 
3.3.1  Addition
 
Theoremadd12 7791 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))
 
Theoremadd32 7792 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))
 
Theoremadd32r 7793 Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 + 𝐶)) = ((𝐴 + 𝐶) + 𝐵))
 
Theoremadd4 7794 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)))
 
Theoremadd42 7795 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)))
 
Theoremadd12i 7796 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))
 
Theoremadd32i 7797 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵)
 
Theoremadd4i 7798 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ       ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))
 
Theoremadd42i 7799 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ       ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))
 
Theoremadd12d 7800 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12832
  Copyright terms: Public domain < Previous  Next >