P. 119 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 11801-11900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	divmul2 11801	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 11802	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 11803	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 11804	Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ)
Theorem	divcan2 11805	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 11806	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 11807	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 11808	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 11809	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 11810	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 11811	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 11812	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 11813	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 11814	Relationship between division and reciprocal. (Contributed by NM, 7-Feb-2006.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴))
Theorem	divass 11815	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 11816	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 11817	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 11818	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 11819	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 11820	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 11821	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 11822	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 11823	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 11824	A cancellation law for division. (Contributed by NM, 8-Feb-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
Theorem	div11 11825	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	div11OLD 11826	Obsolete version of div11 11825 as of 9-Jul-2025. (Contributed by NM, 20-Apr-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵))
Theorem	diveq1 11827	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 11828	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	dividOLD 11829	Obsolete version of divid 11828 as of 9-Jul-2025. (Contributed by NM, 1-Aug-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴 / 𝐴) = 1)
Theorem	div0 11830	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	div0OLD 11831	Obsolete version of div0 11830 as of 9-Jul-2025. (Contributed by NM, 14-Mar-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0 / 𝐴) = 0)
Theorem	div1 11832	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 11833	1 divided by 1 is 1. (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ (1 / 1) = 1
Theorem	divneg 11834	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 11835	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 11836	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	subdivcomb1 11837	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 11838	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 11839	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 11840	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 11841	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 11842	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 11843	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 11844	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 11845	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 11846	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 11847	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 11848	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 11849	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 11850	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 11851	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 11852	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 11853	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 11854	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 11855	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 11856	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 11857	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 11858	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 11859	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 11860	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 11861	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 11862	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 11863	A complex number equals its negative iff it is zero. Deduction form of eqneg 11862. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 = -𝐴 ↔ 𝐴 = 0))
Theorem	eqnegad 11864	If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 11862. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 = -𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = 0)
Theorem	div2neg 11865	Quotient of two negatives. (Contributed by Paul Chapman, 10-Nov-2012.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵))
Theorem	divneg2 11866	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 11867	Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 ≠ 0 → (1 / 𝐴) ∈ ℂ)
Theorem	recne0zi 11868	The reciprocal of a nonzero number is nonzero. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 ≠ 0 → (1 / 𝐴) ≠ 0)
Theorem	recidzi 11869	Multiplication of a number and its reciprocal. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 ≠ 0 → (𝐴 · (1 / 𝐴)) = 1)
Theorem	div1i 11870	A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 / 1) = 𝐴
Theorem	eqnegi 11871	A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 = -𝐴 ↔ 𝐴 = 0)
Theorem	reccli 11872	Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    ⇒   ⊢ (1 / 𝐴) ∈ ℂ
Theorem	recidi 11873	Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    ⇒   ⊢ (𝐴 · (1 / 𝐴)) = 1
Theorem	recreci 11874	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 11875	A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    ⇒   ⊢ (𝐴 / 𝐴) = 1
Theorem	div0i 11876	Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    ⇒   ⊢ (0 / 𝐴) = 0
Theorem	divclzi 11877	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 11878	A cancellation law for division. (Contributed by NM, 2-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
Theorem	divcan2zi 11879	A cancellation law for division. (Contributed by NM, 10-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
Theorem	divreczi 11880	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 11881	A cancellation law for division. (Eliminates a hypothesis of divcan3i 11888 with the weak deduction theorem.) (Contributed by NM, 3-Feb-2004.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
Theorem	divcan4zi 11882	A cancellation law for division. (Contributed by NM, 12-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
Theorem	rec11i 11883	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 11884	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 11885	A cancellation law for division. (Contributed by NM, 9-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ (𝐵 · (𝐴 / 𝐵)) = 𝐴
Theorem	divcan1i 11886	A cancellation law for division. (Contributed by NM, 18-May-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ ((𝐴 / 𝐵) · 𝐵) = 𝐴
Theorem	divreci 11887	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 11888	A cancellation law for division. (Contributed by NM, 16-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ ((𝐵 · 𝐴) / 𝐵) = 𝐴
Theorem	divcan4i 11889	A cancellation law for division. (Contributed by NM, 18-May-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐵 ≠ 0    ⇒   ⊢ ((𝐴 · 𝐵) / 𝐵) = 𝐴
Theorem	divne0i 11890	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 11891	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 11892	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 11893	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 11894	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 11895	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 11896	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 11897	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 11898	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))
Theorem	divdiri 11899	Distribution of division over addition. (Contributed by NM, 16-Feb-1995.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐶 ≠ 0    ⇒   ⊢ ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶))
Theorem	div23i 11900	A commutative/associative law for division. (Contributed by NM, 3-Sep-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐶 ≠ 0    ⇒   ⊢ ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)