P. 115 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 11401-11500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	leloed 11401	'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))
Theorem	eqleltd 11402	Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ ¬ 𝐴 < 𝐵)))
Theorem	ltlend 11403	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴)))
Theorem	lenltd 11404	'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
Theorem	ltnled 11405	'Less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
Theorem	ltled 11406	'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐵)
Theorem	ltnsymd 11407	'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐵 < 𝐴)
Theorem	nltled 11408	'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ¬ 𝐵 < 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐵)
Theorem	lensymd 11409	'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐵 < 𝐴)
Theorem	letrid 11410	Trichotomy law for 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
Theorem	leltned 11411	'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴))
Theorem	leneltd 11412	'Less than or equal to' and 'not equals' implies 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 < 𝐵)
Theorem	mulgt0d 11413	The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ (𝜑 → 0 < 𝐵)    ⇒   ⊢ (𝜑 → 0 < (𝐴 · 𝐵))
Theorem	ltadd2d 11414	Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)))
Theorem	letrd 11415	Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐶)
Theorem	lelttrd 11416	Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 < 𝐶)
Theorem	ltadd2dd 11417	Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 + 𝐴) < (𝐶 + 𝐵))
Theorem	ltletrd 11418	Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 < 𝐶)
Theorem	lttrd 11419	Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 < 𝐶)
Theorem	lelttrdi 11420	If a number is less than another number, and the other number is less than or equal to a third number, the first number is less than the third number. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ))    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐴 < 𝐶))
Theorem	dedekind 11421*	The Dedekind cut theorem. This theorem, which may be used to replace ax-pre-sup 11230 with appropriate adjustments, states that, if 𝐴 completely preceeds 𝐵, then there is some number separating the two of them. (Contributed by Scott Fenton, 13-Jun-2013.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))
Theorem	dedekindle 11422*	The Dedekind cut theorem, with the hypothesis weakened to only require non-strict less than. (Contributed by Scott Fenton, 2-Jul-2013.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))
5.2.5  Initial properties of the complex numbers
Theorem	mul12 11423	Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
Theorem	mul32 11424	Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))
Theorem	mul31 11425	Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))
Theorem	mul4 11426	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)))
Theorem	mul4r 11427	Rearrangement of 4 factors: swap the right factors in the factors of a product of two products. (Contributed by AV, 4-Mar-2023.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐷) · (𝐶 · 𝐵)))
Theorem	muladd11 11428	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))))
Theorem	1p1times 11429	Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴))
Theorem	peano2cn 11430	A theorem for complex numbers analogous the second Peano postulate peano2nn 12275. (Contributed by NM, 17-Aug-2005.)
⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)
Theorem	peano2re 11431	A theorem for reals analogous the second Peano postulate peano2nn 12275. (Contributed by NM, 5-Jul-2005.)
⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)
Theorem	readdcan 11432	Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) ↔ 𝐴 = 𝐵))
Theorem	00id 11433	0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
⊢ (0 + 0) = 0
Theorem	mul02lem1 11434	Lemma for mul02 11436. If any real does not produce 0 when multiplied by 0, then any complex is equal to double itself. (Contributed by Scott Fenton, 3-Jan-2013.)
⊢ (((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) ∧ 𝐵 ∈ ℂ) → 𝐵 = (𝐵 + 𝐵))
Theorem	mul02lem2 11435	Lemma for mul02 11436. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.)
⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0)
Theorem	mul02 11436	Multiplication by 0. Theorem I.6 of [Apostol] p. 18. Based on ideas by Eric Schmidt. (Contributed by NM, 10-Aug-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0)
Theorem	mul01 11437	Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
Theorem	addrid 11438	0 is an additive identity. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
Theorem	cnegex 11439*	Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
Theorem	cnegex2 11440*	Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0)
Theorem	addlid 11441	0 is a left identity for addition. This used to be one of our complex number axioms, until it was discovered that it was dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
Theorem	addcan 11442	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))
Theorem	addcan2 11443	Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	addcom 11444	Addition commutes. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	addridi 11445	0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 + 0) = 𝐴
Theorem	addlidi 11446	0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (0 + 𝐴) = 𝐴
Theorem	mul02i 11447	Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (0 · 𝐴) = 0
Theorem	mul01i 11448	Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 0) = 0
Theorem	addcomi 11449	Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴)
Theorem	addcomli 11450	Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 + 𝐵) = 𝐶    ⇒   ⊢ (𝐵 + 𝐴) = 𝐶
Theorem	addcani 11451	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)
Theorem	addcan2i 11452	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵)
Theorem	mul12i 11453	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))
Theorem	mul32i 11454	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)
Theorem	mul4i 11455	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))
Theorem	mul02d 11456	Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (0 · 𝐴) = 0)
Theorem	mul01d 11457	Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 0) = 0)
Theorem	addridd 11458	0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 + 0) = 𝐴)
Theorem	addlidd 11459	0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (0 + 𝐴) = 𝐴)
Theorem	addcomd 11460	Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	addcand 11461	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))
Theorem	addcan2d 11462	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	addcanad 11463	Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 11461. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐴 + 𝐶))    ⇒   ⊢ (𝜑 → 𝐵 = 𝐶)
Theorem	addcan2ad 11464	Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 11462. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	addneintrd 11465	Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 11463. Consequence of addcand 11461. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ≠ (𝐴 + 𝐶))
Theorem	addneintr2d 11466	Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 11464. Consequence of addcan2d 11462. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐶) ≠ (𝐵 + 𝐶))
Theorem	mul12d 11467	Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
Theorem	mul32d 11468	Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))
Theorem	mul31d 11469	Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))
Theorem	mul4d 11470	Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷)))
Theorem	muladd11r 11471	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))
Theorem	comraddd 11472	Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐵))
Theorem	comraddi 11473	Commute RHS addition. See addcomli 11450 to commute addition on LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐴 = (𝐵 + 𝐶)    ⇒   ⊢ 𝐴 = (𝐶 + 𝐵)
Theorem	ltaddneg 11474	Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))
Theorem	ltaddnegr 11475	Adding a negative number to another number decreases it. (Contributed by AV, 19-Mar-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐴 + 𝐵) < 𝐵))
5.3  Real and complex numbers - basic operations
5.3.1  Addition
Theorem	add12 11476	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)))
Theorem	add32 11477	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))
Theorem	add32r 11478	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))
Theorem	add4 11479	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)))
Theorem	add42 11480	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)))
Theorem	add12i 11481	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))
Theorem	add32i 11482	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)
Theorem	add4i 11483	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))
Theorem	add42i 11484	Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐷 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))
Theorem	add12d 11485	Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))
Theorem	add32d 11486	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))
Theorem	add4d 11487	Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)))
Theorem	add42d 11488	Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)))
5.3.2  Subtraction
Syntax	cmin 11489	Extend class notation to include subtraction.
class −
Syntax	cneg 11490	Extend class notation to include unary minus. The symbol - is not a class by itself but part of a compound class definition. We do this rather than making it a formal function since it is so commonly used. Note: We use different symbols for unary minus (-) and subtraction cmin 11489 (−) to prevent syntax ambiguity. For example, looking at the syntax definition co 7430, if we used the same symbol then "( − 𝐴 − 𝐵) " could mean either "− 𝐴 " minus "𝐵", or it could represent the (meaningless) operation of classes "− " and "− 𝐵 " connected with "operation" "𝐴". On the other hand, "(-𝐴 − 𝐵) " is unambiguous.
class -𝐴
Definition	df-sub 11491*	Define subtraction. Theorem subval 11496 shows its value (and describes how this definition works), Theorem subaddi 11593 relates it to addition, and Theorems subcli 11582 and resubcli 11568 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
Definition	df-neg 11492	Define the negative of a number (unary minus). We use different symbols for unary minus (-) and subtraction (−) to prevent syntax ambiguity. See cneg 11490 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
⊢ -𝐴 = (0 − 𝐴)
Theorem	0cnALT 11493	Alternate proof of 0cn 11250 which does not reference ax-1cn 11210. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ∈ ℂ
Theorem	0cnALT2 11494	Alternate proof of 0cnALT 11493 which is shorter, but depends on ax-8 2107, ax-13 2374, ax-sep 5301, ax-nul 5311, ax-pow 5370, ax-pr 5437, ax-un 7753, and every complex number axiom except ax-pre-mulgt0 11229 and ax-pre-sup 11230. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ∈ ℂ
Theorem	negeu 11495*	Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
Theorem	subval 11496*	Value of subtraction, which is the (unique) element 𝑥 such that 𝐵 + 𝑥 = 𝐴. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
Theorem	negeq 11497	Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵)
Theorem	negeqi 11498	Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ -𝐴 = -𝐵
Theorem	negeqd 11499	Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → -𝐴 = -𝐵)
Theorem	nfnegd 11500	Deduction version of nfneg 11501. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥-𝐴)