P. 121 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12001-12100   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-div 12001*	Define division. Theorem divmuli 12098 relates it to multiplication, and divcli 12086 and redivcli 12111 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divval 12003 instead. (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
Theorem	1div0 12002	You can't divide by zero, because division explicitly excludes zero from the domain of the function. Thus, by the definition of function value, it evaluates to the empty set. (This theorem is for information only and normally is not referenced by other proofs. To be meaningful, it assumes that ∅ is not a complex number, which depends on the particular complex number construction that is used.) (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof shortened by SN, 5-Jun-2025.) (New usage is discouraged.)
⊢ (1 / 0) = ∅
Theorem	divval 12003*	Value of division: if 𝐴 and 𝐵 are complex numbers with 𝐵 nonzero, then (𝐴 / 𝐵) is the (unique) complex number such that (𝐵 · 𝑥) = 𝐴. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
Theorem	divmul 12004	Relationship between division and multiplication. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 17-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ (𝐶 · 𝐵) = 𝐴))
Theorem	divmul2 12005	Relationship between division and multiplication. (Contributed by NM, 7-Feb-2006.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐶 · 𝐵)))
Theorem	divmul3 12006	Relationship between division and multiplication. (Contributed by NM, 13-Feb-2006.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐵 · 𝐶)))
Theorem	divcl 12007	Closure law for division. (Contributed by NM, 21-Jul-2001.) (Proof shortened by Mario Carneiro, 17-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℂ)
Theorem	reccl 12008	Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ)
Theorem	divcan2 12009	A cancellation law for division. (Contributed by NM, 3-Feb-2004.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
Theorem	divcan1 12010	A cancellation law for division. (Contributed by NM, 5-Jun-2004.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
Theorem	diveq0 12011	A ratio is zero iff the numerator is zero. (Contributed by NM, 20-Apr-2006.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0))
Theorem	divne0b 12012	The ratio of nonzero numbers is nonzero. (Contributed by NM, 2-Aug-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 ≠ 0 ↔ (𝐴 / 𝐵) ≠ 0))
Theorem	divne0 12013	The ratio of nonzero numbers is nonzero. (Contributed by NM, 28-Dec-2007.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ≠ 0)
Theorem	recne0 12014	The reciprocal of a nonzero number is nonzero. (Contributed by NM, 9-Feb-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ≠ 0)
Theorem	recid 12015	Multiplication of a number and its reciprocal. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴 · (1 / 𝐴)) = 1)
Theorem	recid2 12016	Multiplication of a number and its reciprocal. (Contributed by NM, 22-Jun-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((1 / 𝐴) · 𝐴) = 1)
Theorem	divrec 12017	Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
Theorem	divrec2 12018	Relationship between division and reciprocal. (Contributed by NM, 7-Feb-2006.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴))
Theorem	divass 12019	An associative law for division. (Contributed by NM, 2-Aug-2004.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)))
Theorem	div23 12020	A commutative/associative law for division. (Contributed by NM, 2-Aug-2004.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵))
Theorem	div32 12021	A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = (𝐴 · (𝐶 / 𝐵)))
Theorem	div13 12022	A commutative/associative law for division. (Contributed by NM, 22-Apr-2005.)
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = ((𝐶 / 𝐵) · 𝐴))
Theorem	div12 12023	A commutative/associative law for division. (Contributed by NM, 30-Apr-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶)))
Theorem	divmulass 12024	An associative law for division and multiplication. (Contributed by AV, 10-Jul-2021.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = ((𝐴 · 𝐵) · (𝐶 / 𝐷)))
Theorem	divmulasscom 12025	An associative/commutative law for division and multiplication. (Contributed by AV, 10-Jul-2021.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = (𝐵 · ((𝐴 · 𝐶) / 𝐷)))
Theorem	divdir 12026	Distribution of division over addition. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))
Theorem	divcan3 12027	A cancellation law for division. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
Theorem	divcan4 12028	A cancellation law for division. (Contributed by NM, 8-Feb-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
Theorem	div11 12029	One-to-one relationship for division. (Contributed by NM, 20-Apr-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.) (Proof shortened by SN, 9-Jul-2025.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵))
Theorem	diveq1 12030	Equality in terms of unit ratio. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵))
Theorem	divid 12031	A number divided by itself is one. (Contributed by NM, 1-Aug-2004.) (Proof shortened by SN, 9-Jul-2025.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴 / 𝐴) = 1)
Theorem	div0 12032	Division into zero is zero. (Contributed by NM, 14-Mar-2005.) (Proof shortened by SN, 9-Jul-2025.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0 / 𝐴) = 0)
Theorem	div1 12033	A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴)
Theorem	1div1e1 12034	1 divided by 1 is 1. (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ (1 / 1) = 1
Theorem	divneg 12035	Move negative sign inside of a division. (Contributed by NM, 17-Sep-2004.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → -(𝐴 / 𝐵) = ( -𝐴 / 𝐵))
Theorem	muldivdir 12036	Distribution of division over addition with a multiplication. (Contributed by AV, 1-Jul-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((𝐶 · 𝐴) + 𝐵) / 𝐶) = (𝐴 + (𝐵 / 𝐶)))
Theorem	divsubdir 12037	Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 − 𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶)))
Theorem	muldivdid 12038	Distribution of division over addition with a multiplication. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → (((𝐴 · 𝐵) + 𝐶) / 𝐵) = (𝐴 + (𝐶 / 𝐵)))
Theorem	subdivcomb1 12039	Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((𝐶 · 𝐴) − 𝐵) / 𝐶) = (𝐴 − (𝐵 / 𝐶)))
Theorem	subdivcomb2 12040	Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 − (𝐶 · 𝐵)) / 𝐶) = ((𝐴 / 𝐶) − 𝐵))
Theorem	recrec 12041	A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 26-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / (1 / 𝐴)) = 𝐴)
Theorem	rec11 12042	Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵))
Theorem	rec11r 12043	Mutual reciprocals. (Contributed by Paul Chapman, 18-Oct-2007.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((1 / 𝐴) = 𝐵 ↔ (1 / 𝐵) = 𝐴))
Theorem	divmuldiv 12044	Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by NM, 1-Aug-2004.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐴 · 𝐵) / (𝐶 · 𝐷)))
Theorem	divdivdiv 12045	Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by NM, 2-Aug-2004.)
⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶)))
Theorem	divcan5 12046	Cancellation of common factor in a ratio. (Contributed by NM, 9-Jan-2006.)
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵))
Theorem	divmul13 12047	Swap the denominators in the product of two ratios. (Contributed by NM, 3-May-2005.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐵 / 𝐶) · (𝐴 / 𝐷)))
Theorem	divmul24 12048	Swap the numerators in the product of two ratios. (Contributed by NM, 3-May-2005.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐴 / 𝐷) · (𝐵 / 𝐶)))
Theorem	divmuleq 12049	Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 23-Feb-2014.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐶) = (𝐵 / 𝐷) ↔ (𝐴 · 𝐷) = (𝐵 · 𝐶)))
Theorem	recdiv 12050	The reciprocal of a ratio. (Contributed by NM, 3-Aug-2004.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴))
Theorem	divcan6 12051	Cancellation of inverted fractions. (Contributed by NM, 28-Dec-2007.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1)
Theorem	divdiv32 12052	Swap denominators in a division. (Contributed by NM, 2-Aug-2004.)
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵))
Theorem	divcan7 12053	Cancel equal divisors in a division. (Contributed by Jeff Hankins, 29-Sep-2013.)
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) / (𝐵 / 𝐶)) = (𝐴 / 𝐵))
Theorem	dmdcan 12054	Cancellation law for division and multiplication. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · (𝐶 / 𝐴)) = (𝐶 / 𝐵))
Theorem	divdiv1 12055	Division into a fraction. (Contributed by NM, 31-Dec-2007.)
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶)))
Theorem	divdiv2 12056	Division by a fraction. (Contributed by NM, 27-Dec-2008.)
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 / (𝐵 / 𝐶)) = ((𝐴 · 𝐶) / 𝐵))
Theorem	recdiv2 12057	Division into a reciprocal. (Contributed by NM, 19-Oct-2007.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵)))
Theorem	ddcan 12058	Cancellation in a double division. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 / (𝐴 / 𝐵)) = 𝐵)
Theorem	divadddiv 12059	Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 2-May-2016.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐶) + (𝐵 / 𝐷)) = (((𝐴 · 𝐷) + (𝐵 · 𝐶)) / (𝐶 · 𝐷)))
Theorem	divsubdiv 12060	Subtraction of two ratios. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 2-May-2016.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐶) − (𝐵 / 𝐷)) = (((𝐴 · 𝐷) − (𝐵 · 𝐶)) / (𝐶 · 𝐷)))
Theorem	conjmul 12061	Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 12-Nov-2006.)
⊢ (((𝑃 ∈ ℂ ∧ 𝑃 ≠ 0) ∧ (𝑄 ∈ ℂ ∧ 𝑄 ≠ 0)) → (((1 / 𝑃) + (1 / 𝑄)) = 1 ↔ ((𝑃 − 1) · (𝑄 − 1)) = 1))
Theorem	rereccl 12062	Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℝ)
Theorem	redivcl 12063	Closure law for division of reals. (Contributed by NM, 27-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℝ)
Theorem	eqneg 12064	A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℂ → (𝐴 = -𝐴 ↔ 𝐴 = 0))
Theorem	eqnegd 12065	A complex number equals its negative iff it is zero. Deduction form of eqneg 12064. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 = -𝐴 ↔ 𝐴 = 0))
Theorem	eqnegad 12066	If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 12064. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 = -𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = 0)
Theorem	div2neg 12067	Quotient of two negatives. (Contributed by Paul Chapman, 10-Nov-2012.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ( -𝐴 / -𝐵) = (𝐴 / 𝐵))
Theorem	divneg2 12068	Move negative sign inside of a division. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → -(𝐴 / 𝐵) = (𝐴 / -𝐵))
Theorem	recclzi 12069	Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 ≠ 0 → (1 / 𝐴) ∈ ℂ)
Theorem	recne0zi 12070	The reciprocal of a nonzero number is nonzero. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 ≠ 0 → (1 / 𝐴) ≠ 0)
Theorem	recidzi 12071	Multiplication of a number and its reciprocal. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 ≠ 0 → (𝐴 · (1 / 𝐴)) = 1)
Theorem	div1i 12072	A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 / 1) = 𝐴
Theorem	eqnegi 12073	A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 = -𝐴 ↔ 𝐴 = 0)
Theorem	reccli 12074	Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    ⇒   ⊢ (1 / 𝐴) ∈ ℂ
Theorem	recidi 12075	Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    ⇒   ⊢ (𝐴 · (1 / 𝐴)) = 1
Theorem	recreci 12076	A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    ⇒   ⊢ (1 / (1 / 𝐴)) = 𝐴
Theorem	dividi 12077	A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    ⇒   ⊢ (𝐴 / 𝐴) = 1
Theorem	div0i 12078	Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    ⇒   ⊢ (0 / 𝐴) = 0
Theorem	divclzi 12079	Closure law for division. (Contributed by NM, 7-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → (𝐴 / 𝐵) ∈ ℂ)
Theorem	divcan1zi 12080	A cancellation law for division. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
Theorem	divcan2zi 12081	A cancellation law for division. (Contributed by NM, 10-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
Theorem	divreczi 12082	Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by NM, 11-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
Theorem	divcan3zi 12083	A cancellation law for division. (Eliminates a hypothesis of divcan3i 12090 with the weak deduction theorem.) (Contributed by NM, 3-Feb-2004.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
Theorem	divcan4zi 12084	A cancellation law for division. (Contributed by NM, 12-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
Theorem	rec11i 12085	Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) → ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵))
Theorem	divcli 12086	Closure law for division. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ (𝐴 / 𝐵) ∈ ℂ
Theorem	divcan2i 12087	A cancellation law for division. (Contributed by NM, 9-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ (𝐵 · (𝐴 / 𝐵)) = 𝐴
Theorem	divcan1i 12088	A cancellation law for division. (Contributed by NM, 18-May-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ ((𝐴 / 𝐵) · 𝐵) = 𝐴
Theorem	divreci 12089	Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))
Theorem	divcan3i 12090	A cancellation law for division. (Contributed by NM, 16-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ ((𝐵 · 𝐴) / 𝐵) = 𝐴
Theorem	divcan4i 12091	A cancellation law for division. (Contributed by NM, 18-May-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ ((𝐴 · 𝐵) / 𝐵) = 𝐴
Theorem	divne0i 12092	The ratio of nonzero numbers is nonzero. (Contributed by NM, 9-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ (𝐴 / 𝐵) ≠ 0
Theorem	rec11ii 12093	Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵)
Theorem	divasszi 12094	An associative law for division. (Contributed by NM, 12-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ (𝐶 ≠ 0 → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)))
Theorem	divmulzi 12095	Relationship between division and multiplication. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴))
Theorem	divdirzi 12096	Distribution of division over addition. (Contributed by NM, 31-Jul-2004.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ (𝐶 ≠ 0 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))
Theorem	divdiv23zi 12097	Swap denominators in a division. (Contributed by NM, 15-Sep-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐵 ≠ 0 ∧ 𝐶 ≠ 0) → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵))
Theorem	divmuli 12098	Relationship between division and multiplication. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴)
Theorem	divdiv32i 12099	Swap denominators in a division. (Contributed by NM, 15-Sep-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    &   ⊢ 𝐶 ≠ 0    ⇒   ⊢ ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵)
Theorem	divassi 12100	An associative law for division. (Contributed by NM, 15-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐶 ≠ 0    ⇒   ⊢ ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))