P. 81 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8001-8100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ltnri 8001	'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ ¬ 𝐴 < 𝐴
Theorem	eqlei 8002	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 = 𝐵 → 𝐴 ≤ 𝐵)
Theorem	eqlei2 8003	Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (𝐵 = 𝐴 → 𝐵 ≤ 𝐴)
Theorem	gtneii 8004	'Less than' implies not equal. See also gtapii 8542 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐴 < 𝐵    ⇒   ⊢ 𝐵 ≠ 𝐴
Theorem	ltneii 8005	'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐴 < 𝐵    ⇒   ⊢ 𝐴 ≠ 𝐵
Theorem	lttri3i 8006	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
Theorem	letri3i 8007	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))
Theorem	ltnsymi 8008	'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)
Theorem	lenlti 8009	'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)
Theorem	ltlei 8010	'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)
Theorem	ltleii 8011	'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐴 < 𝐵    ⇒   ⊢ 𝐴 ≤ 𝐵
Theorem	ltnei 8012	'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 < 𝐵 → 𝐵 ≠ 𝐴)
Theorem	lttri 8013	'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)
Theorem	lelttri 8014	'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)
Theorem	ltletri 8015	'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)
Theorem	letri 8016	'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)
Theorem	le2tri3i 8017	Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴))
Theorem	mulgt0i 8018	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))
Theorem	mulgt0ii 8019	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)
Theorem	ltnrd 8020	'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → ¬ 𝐴 < 𝐴)
Theorem	gtned 8021	'Less than' implies not equal. See also gtapd 8545 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ≠ 𝐴)
Theorem	ltned 8022	'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	lttri3d 8023	Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
Theorem	letri3d 8024	Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)))
Theorem	eqleltd 8025	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	lenltd 8026	'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	ltled 8027	'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 8028	'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 8029	'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 8030	'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	mulgt0d 8031	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	letrd 8032	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 8033	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	lttrd 8034	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	0lt1 8035	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
Theorem	ltntri 8036	Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 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))
4.2.5  Initial properties of the complex numbers
Theorem	mul12 8037	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 8038	Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))
Theorem	mul31 8039	Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))
Theorem	mul4 8040	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	muladd11 8041	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 8042	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 8043	A theorem for complex numbers analogous the second Peano postulate peano2 4577. (Contributed by NM, 17-Aug-2005.)
⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)
Theorem	peano2re 8044	A theorem for reals analogous the second Peano postulate peano2 4577. (Contributed by NM, 5-Jul-2005.)
⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)
Theorem	addcom 8045	Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	addid1 8046	0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
Theorem	addid2 8047	0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
Theorem	readdcan 8048	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))
Theorem	00id 8049	0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
⊢ (0 + 0) = 0
Theorem	addid1i 8050	0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 + 0) = 𝐴
Theorem	addid2i 8051	0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (0 + 𝐴) = 𝐴
Theorem	addcomi 8052	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 8053	Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 + 𝐵) = 𝐶    ⇒   ⊢ (𝐵 + 𝐴) = 𝐶
Theorem	mul12i 8054	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 8055	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 8056	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	addid1d 8057	0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 + 0) = 𝐴)
Theorem	addid2d 8058	0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (0 + 𝐴) = 𝐴)
Theorem	addcomd 8059	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	mul12d 8060	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 8061	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 8062	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 8063	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 8064	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 8065	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))
4.3  Real and complex numbers - basic operations
4.3.1  Addition
Theorem	add12 8066	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 8067	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 8068	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 8069	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 8070	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 8071	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 8072	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 8073	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 8074	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))
Theorem	add12d 8075	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 8076	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 8077	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 8078	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)))
4.3.2  Subtraction
Syntax	cmin 8079	Extend class notation to include subtraction.
class −
Syntax	cneg 8080	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 8079 (−) to prevent syntax ambiguity. For example, looking at the syntax definition co 5851, 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 8081*	Define subtraction. Theorem subval 8100 shows its value (and describes how this definition works), Theorem subaddi 8195 relates it to addition, and Theorems subcli 8184 and resubcli 8171 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
Definition	df-neg 8082	Define the negative of a number (unary minus). We use different symbols for unary minus (-) and subtraction (−) to prevent syntax ambiguity. See cneg 8080 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
⊢ -𝐴 = (0 − 𝐴)
Theorem	cnegexlem1 8083	Addition cancellation of a real number from two complex numbers. Lemma for cnegex 8086. (Contributed by Eric Schmidt, 22-May-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))
Theorem	cnegexlem2 8084	Existence of a real number which produces a real number when multiplied by i. (Hint: zero is such a number, although we don't need to prove that yet). Lemma for cnegex 8086. (Contributed by Eric Schmidt, 22-May-2007.)
⊢ ∃𝑦 ∈ ℝ (i · 𝑦) ∈ ℝ
Theorem	cnegexlem3 8085*	Existence of real number difference. Lemma for cnegex 8086. (Contributed by Eric Schmidt, 22-May-2007.)
⊢ ((𝑏 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ∃𝑐 ∈ ℝ (𝑏 + 𝑐) = 𝑦)
Theorem	cnegex 8086*	Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
Theorem	cnegex2 8087*	Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0)
Theorem	addcan 8088	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 8089	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	addcani 8090	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 8091	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	addcand 8092	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 8093	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 8094	Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 8092. (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 8095	Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 8093. (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 8096	Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 8094. Consequence of addcand 8092. (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 8097	Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 8095. Consequence of addcan2d 8093. (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	0cnALT 8098	Alternate proof of 0cn 7901. (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 8099*	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 8100*	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))