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Theorem List for Metamath Proof Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrecid 11301 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)
 
Theoremrecid2 11302 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)
 
Theoremdivrec 11303 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)))
 
Theoremdivrec2 11304 Relationship between division and reciprocal. (Contributed by NM, 7-Feb-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴))
 
Theoremdivass 11305 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)))
 
Theoremdiv23 11306 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))
 
Theoremdiv32 11307 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)))
 
Theoremdiv13 11308 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))
 
Theoremdiv12 11309 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)))
 
Theoremdivmulass 11310 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)))
 
Theoremdivmulasscom 11311 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)))
 
Theoremdivdir 11312 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)))
 
Theoremdivcan3 11313 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)
 
Theoremdivcan4 11314 A cancellation law for division. (Contributed by NM, 8-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
Theoremdiv11 11315 One-to-one relationship for division. (Contributed by NM, 20-Apr-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremdivid 11316 A number divided by itself is one. (Contributed by NM, 1-Aug-2004.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴 / 𝐴) = 1)
 
Theoremdiv0 11317 Division into zero is zero. (Contributed by NM, 14-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0 / 𝐴) = 0)
 
Theoremdiv1 11318 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)
 
Theorem1div1e1 11319 1 divided by 1 is 1. (Contributed by David A. Wheeler, 7-Dec-2018.)
(1 / 1) = 1
 
Theoremdiveq1 11320 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))
 
Theoremdivneg 11321 Move negative sign inside of a division. (Contributed by NM, 17-Sep-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵))
 
Theoremmuldivdir 11322 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)))
 
Theoremdivsubdir 11323 Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶)))
 
Theoremsubdivcomb1 11324 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)))
 
Theoremsubdivcomb2 11325 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))
 
Theoremrecrec 11326 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)
 
Theoremrec11 11327 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))
 
Theoremrec11r 11328 Mutual reciprocals. (Contributed by Paul Chapman, 18-Oct-2007.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((1 / 𝐴) = 𝐵 ↔ (1 / 𝐵) = 𝐴))
 
Theoremdivmuldiv 11329 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)))
 
Theoremdivdivdiv 11330 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)))
 
Theoremdivcan5 11331 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))
 
Theoremdivmul13 11332 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)))
 
Theoremdivmul24 11333 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)))
 
Theoremdivmuleq 11334 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)))
 
Theoremrecdiv 11335 The reciprocal of a ratio. (Contributed by NM, 3-Aug-2004.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴))
 
Theoremdivcan6 11336 Cancellation of inverted fractions. (Contributed by NM, 28-Dec-2007.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1)
 
Theoremdivdiv32 11337 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))
 
Theoremdivcan7 11338 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))
 
Theoremdmdcan 11339 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))
 
Theoremdivdiv1 11340 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)))
 
Theoremdivdiv2 11341 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))
 
Theoremrecdiv2 11342 Division into a reciprocal. (Contributed by NM, 19-Oct-2007.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵)))
 
Theoremddcan 11343 Cancellation in a double division. (Contributed by Mario Carneiro, 26-Apr-2015.)
(((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 / (𝐴 / 𝐵)) = 𝐵)
 
Theoremdivadddiv 11344 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)))
 
Theoremdivsubdiv 11345 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)))
 
Theoremconjmul 11346 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))
 
Theoremrereccl 11347 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℝ)
 
Theoremredivcl 11348 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)
 
Theoremeqneg 11349 A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴 = -𝐴𝐴 = 0))
 
Theoremeqnegd 11350 A complex number equals its negative iff it is zero. Deduction form of eqneg 11349. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 = -𝐴𝐴 = 0))
 
Theoremeqnegad 11351 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 11349. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 = -𝐴)       (𝜑𝐴 = 0)
 
Theoremdiv2neg 11352 Quotient of two negatives. (Contributed by Paul Chapman, 10-Nov-2012.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵))
 
Theoremdivneg2 11353 Move negative sign inside of a division. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → -(𝐴 / 𝐵) = (𝐴 / -𝐵))
 
Theoremrecclzi 11354 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
𝐴 ∈ ℂ       (𝐴 ≠ 0 → (1 / 𝐴) ∈ ℂ)
 
Theoremrecne0zi 11355 The reciprocal of a nonzero number is nonzero. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℂ       (𝐴 ≠ 0 → (1 / 𝐴) ≠ 0)
 
Theoremrecidzi 11356 Multiplication of a number and its reciprocal. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℂ       (𝐴 ≠ 0 → (𝐴 · (1 / 𝐴)) = 1)
 
Theoremdiv1i 11357 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
𝐴 ∈ ℂ       (𝐴 / 1) = 𝐴
 
Theoremeqnegi 11358 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 = -𝐴𝐴 = 0)
 
Theoremreccli 11359 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       (1 / 𝐴) ∈ ℂ
 
Theoremrecidi 11360 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       (𝐴 · (1 / 𝐴)) = 1
 
Theoremrecreci 11361 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
 
Theoremdividi 11362 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       (𝐴 / 𝐴) = 1
 
Theoremdiv0i 11363 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       (0 / 𝐴) = 0
 
Theoremdivclzi 11364 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)
 
Theoremdivcan1zi 11365 A cancellation law for division. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 ≠ 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
 
Theoremdivcan2zi 11366 A cancellation law for division. (Contributed by NM, 10-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 ≠ 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
 
Theoremdivreczi 11367 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)))
 
Theoremdivcan3zi 11368 A cancellation law for division. (Eliminates a hypothesis of divcan3i 11375 with the weak deduction theorem.) (Contributed by NM, 3-Feb-2004.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 ≠ 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
 
Theoremdivcan4zi 11369 A cancellation law for division. (Contributed by NM, 12-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 ≠ 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
Theoremrec11i 11370 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))
 
Theoremdivcli 11371 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
 
Theoremdivcan2i 11372 A cancellation law for division. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 ≠ 0       (𝐵 · (𝐴 / 𝐵)) = 𝐴
 
Theoremdivcan1i 11373 A cancellation law for division. (Contributed by NM, 18-May-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 ≠ 0       ((𝐴 / 𝐵) · 𝐵) = 𝐴
 
Theoremdivreci 11374 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))
 
Theoremdivcan3i 11375 A cancellation law for division. (Contributed by NM, 16-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 ≠ 0       ((𝐵 · 𝐴) / 𝐵) = 𝐴
 
Theoremdivcan4i 11376 A cancellation law for division. (Contributed by NM, 18-May-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 ≠ 0       ((𝐴 · 𝐵) / 𝐵) = 𝐴
 
Theoremdivne0i 11377 The ratio of nonzero numbers is nonzero. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 ≠ 0    &   𝐵 ≠ 0       (𝐴 / 𝐵) ≠ 0
 
Theoremrec11ii 11378 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)
 
Theoremdivasszi 11379 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)))
 
Theoremdivmulzi 11380 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))
 
Theoremdivdirzi 11381 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)))
 
Theoremdivdiv23zi 11382 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))
 
Theoremdivmuli 11383 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)
 
Theoremdivdiv32i 11384 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)
 
Theoremdivassi 11385 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))
 
Theoremdivdiri 11386 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))
 
Theoremdiv23i 11387 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)
 
Theoremdiv11i 11388 One-to-one relationship for division. (Contributed by NM, 20-Aug-2001.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 ≠ 0       ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵)
 
Theoremdivmuldivi 11389 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by NM, 16-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 ≠ 0    &   𝐷 ≠ 0       ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷))
 
Theoremdivmul13i 11390 Swap denominators of two ratios. (Contributed by NM, 6-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 ≠ 0    &   𝐷 ≠ 0       ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐶 / 𝐵) · (𝐴 / 𝐷))
 
Theoremdivadddivi 11391 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by NM, 21-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 ≠ 0    &   𝐷 ≠ 0       ((𝐴 / 𝐵) + (𝐶 / 𝐷)) = (((𝐴 · 𝐷) + (𝐶 · 𝐵)) / (𝐵 · 𝐷))
 
Theoremdivdivdivi 11392 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by NM, 22-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 ≠ 0    &   𝐷 ≠ 0    &   𝐶 ≠ 0       ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶))
 
Theoremrerecclzi 11393 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
𝐴 ∈ ℝ       (𝐴 ≠ 0 → (1 / 𝐴) ∈ ℝ)
 
Theoremrereccli 11394 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
𝐴 ∈ ℝ    &   𝐴 ≠ 0       (1 / 𝐴) ∈ ℝ
 
Theoremredivclzi 11395 Closure law for division of reals. (Contributed by NM, 9-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐵 ≠ 0 → (𝐴 / 𝐵) ∈ ℝ)
 
Theoremredivcli 11396 Closure law for division of reals. (Contributed by NM, 9-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐵 ≠ 0       (𝐴 / 𝐵) ∈ ℝ
 
Theoremdiv1d 11397 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 / 1) = 𝐴)
 
Theoremreccld 11398 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (1 / 𝐴) ∈ ℂ)
 
Theoremrecne0d 11399 The reciprocal of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (1 / 𝐴) ≠ 0)
 
Theoremrecidd 11400 Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (𝐴 · (1 / 𝐴)) = 1)
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