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Theorem List for Metamath Proof Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrecidzi 10501 Multiplication of a number and its reciprocal. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℂ       (𝐴 ≠ 0 → (𝐴 · (1 / 𝐴)) = 1)
 
Theoremdiv1i 10502 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
𝐴 ∈ ℂ       (𝐴 / 1) = 𝐴
 
Theoremeqnegi 10503 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 = -𝐴𝐴 = 0)
 
Theoremreccli 10504 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       (1 / 𝐴) ∈ ℂ
 
Theoremrecidi 10505 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       (𝐴 · (1 / 𝐴)) = 1
 
Theoremrecreci 10506 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 10507 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       (𝐴 / 𝐴) = 1
 
Theoremdiv0i 10508 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       (0 / 𝐴) = 0
 
Theoremdivclzi 10509 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 10510 A cancellation law for division. (Contributed by NM, 2-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 ≠ 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
 
Theoremdivcan2zi 10511 A cancellation law for division. (Contributed by NM, 10-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 ≠ 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
 
Theoremdivreczi 10512 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 10513 A cancellation law for division. (Eliminates a hypothesis of divcan3i 10520 with the weak deduction theorem.) (Contributed by NM, 3-Feb-2004.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 ≠ 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
 
Theoremdivcan4zi 10514 A cancellation law for division. (Contributed by NM, 12-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 ≠ 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
Theoremrec11i 10515 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 10516 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 10517 A cancellation law for division. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 ≠ 0       (𝐵 · (𝐴 / 𝐵)) = 𝐴
 
Theoremdivcan1i 10518 A cancellation law for division. (Contributed by NM, 18-May-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 ≠ 0       ((𝐴 / 𝐵) · 𝐵) = 𝐴
 
Theoremdivreci 10519 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 10520 A cancellation law for division. (Contributed by NM, 16-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 ≠ 0       ((𝐵 · 𝐴) / 𝐵) = 𝐴
 
Theoremdivcan4i 10521 A cancellation law for division. (Contributed by NM, 18-May-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 ≠ 0       ((𝐴 · 𝐵) / 𝐵) = 𝐴
 
Theoremdivne0i 10522 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 10523 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 10524 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 10525 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 10526 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 10527 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 10528 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 10529 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 10530 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 10531 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 10532 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 10533 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 10534 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 10535 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 10536 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 10537 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 10538 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
𝐴 ∈ ℝ       (𝐴 ≠ 0 → (1 / 𝐴) ∈ ℝ)
 
Theoremrereccli 10539 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
𝐴 ∈ ℝ    &   𝐴 ≠ 0       (1 / 𝐴) ∈ ℝ
 
Theoremredivclzi 10540 Closure law for division of reals. (Contributed by NM, 9-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐵 ≠ 0 → (𝐴 / 𝐵) ∈ ℝ)
 
Theoremredivcli 10541 Closure law for division of reals. (Contributed by NM, 9-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐵 ≠ 0       (𝐴 / 𝐵) ∈ ℝ
 
Theoremdiv1d 10542 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 / 1) = 𝐴)
 
Theoremreccld 10543 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (1 / 𝐴) ∈ ℂ)
 
Theoremrecne0d 10544 The reciprocal of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (1 / 𝐴) ≠ 0)
 
Theoremrecidd 10545 Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (𝐴 · (1 / 𝐴)) = 1)
 
Theoremrecid2d 10546 Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → ((1 / 𝐴) · 𝐴) = 1)
 
Theoremrecrecd 10547 A number is equal to the reciprocal of its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (1 / (1 / 𝐴)) = 𝐴)
 
Theoremdividd 10548 A number divided by itself is one. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (𝐴 / 𝐴) = 1)
 
Theoremdiv0d 10549 Division into zero is zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → (0 / 𝐴) = 0)
 
Theoremdivcld 10550 Closure law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 / 𝐵) ∈ ℂ)
 
Theoremdivcan1d 10551 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
 
Theoremdivcan2d 10552 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
 
Theoremdivrecd 10553 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
 
Theoremdivrec2d 10554 Relationship between division and reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴))
 
Theoremdivcan3d 10555 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
 
Theoremdivcan4d 10556 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
Theoremdiveq0d 10557 A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑 → (𝐴 / 𝐵) = 0)       (𝜑𝐴 = 0)
 
Theoremdiveq1d 10558 Equality in terms of unit ratio. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑 → (𝐴 / 𝐵) = 1)       (𝜑𝐴 = 𝐵)
 
Theoremdiveq1ad 10559 The quotient of two complex numbers is one iff they are equal. Deduction form of diveq1 10467. Generalization of diveq1d 10558. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵))
 
Theoremdiveq0ad 10560 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveq0 10444. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0))
 
Theoremdivne1d 10561 If two complex numbers are unequal, their quotient is not one. Contrapositive of diveq1d 10558. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 / 𝐵) ≠ 1)
 
Theoremdivne0bd 10562 A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 ≠ 0 ↔ (𝐴 / 𝐵) ≠ 0))
 
Theoremdivnegd 10563 Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → -(𝐴 / 𝐵) = (-𝐴 / 𝐵))
 
Theoremdivneg2d 10564 Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → -(𝐴 / 𝐵) = (𝐴 / -𝐵))
 
Theoremdiv2negd 10565 Quotient of two negatives. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (-𝐴 / -𝐵) = (𝐴 / 𝐵))
 
Theoremdivne0d 10566 The ratio of nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 / 𝐵) ≠ 0)
 
Theoremrecdivd 10567 The reciprocal of a ratio. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴))
 
Theoremrecdiv2d 10568 Division into a reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵)))
 
Theoremdivcan6d 10569 Cancellation of inverted fractions. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1)
 
Theoremddcand 10570 Cancellation in a double division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 / (𝐴 / 𝐵)) = 𝐵)
 
Theoremrec11d 10571 Reciprocal is one-to-one. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)    &   (𝜑 → (1 / 𝐴) = (1 / 𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremdivmuld 10572 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴))
 
Theoremdiv32d 10573 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · 𝐶) = (𝐴 · (𝐶 / 𝐵)))
 
Theoremdiv13d 10574 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · 𝐶) = ((𝐶 / 𝐵) · 𝐴))
 
Theoremdivdiv32d 10575 Swap denominators in a division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵))
 
Theoremdivcan5d 10576 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵))
 
Theoremdivcan5rd 10577 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 · 𝐶) / (𝐵 · 𝐶)) = (𝐴 / 𝐵))
 
Theoremdivcan7d 10578 Cancel equal divisors in a division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐶) / (𝐵 / 𝐶)) = (𝐴 / 𝐵))
 
Theoremdmdcand 10579 Cancellation law for division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐵 / 𝐶) · (𝐴 / 𝐵)) = (𝐴 / 𝐶))
 
Theoremdmdcan2d 10580 Cancellation law for division and multiplication. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐵 / 𝐶)) = (𝐴 / 𝐶))
 
Theoremdivdiv1d 10581 Division into a fraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶)))
 
Theoremdivdiv2d 10582 Division by a fraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → (𝐴 / (𝐵 / 𝐶)) = ((𝐴 · 𝐶) / 𝐵))
 
Theoremdivmul2d 10583 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐶) = 𝐵𝐴 = (𝐶 · 𝐵)))
 
Theoremdivmul3d 10584 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐶) = 𝐵𝐴 = (𝐵 · 𝐶)))
 
Theoremdivassd 10585 An associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)))
 
Theoremdiv12d 10586 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶)))
 
Theoremdiv23d 10587 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵))
 
Theoremdivdird 10588 Distribution of division over addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))
 
Theoremdivsubdird 10589 Distribution of division over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶)))
 
Theoremdiv11d 10590 One-to-one relationship for division. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)    &   (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶))       (𝜑𝐴 = 𝐵)
 
Theoremdivmuldivd 10591 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷)))
 
Theoremdivmul13d 10592 Swap denominators of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐶 / 𝐵) · (𝐴 / 𝐷)))
 
Theoremdivmul24d 10593 Swap the numerators in the product of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 / 𝐷) · (𝐶 / 𝐵)))
 
Theoremdivadddivd 10594 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) + (𝐶 / 𝐷)) = (((𝐴 · 𝐷) + (𝐶 · 𝐵)) / (𝐵 · 𝐷)))
 
Theoremdivsubdivd 10595 Subtraction of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) − (𝐶 / 𝐷)) = (((𝐴 · 𝐷) − (𝐶 · 𝐵)) / (𝐵 · 𝐷)))
 
Theoremdivmuleqd 10596 Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) = (𝐶 / 𝐷) ↔ (𝐴 · 𝐷) = (𝐶 · 𝐵)))
 
Theoremdivdivdivd 10597 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐷 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶)))
 
Theoremdiveq1bd 10598 If two complex numbers are equal, their quotient is one. One-way deduction form of diveq1 10467. Converse of diveq1d 10558. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 / 𝐵) = 1)
 
Theoremdiv2sub 10599 Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐶𝐷)) → ((𝐴𝐵) / (𝐶𝐷)) = ((𝐵𝐴) / (𝐷𝐶)))
 
Theoremdiv2subd 10600 Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2sub 10599. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐶𝐷)       (𝜑 → ((𝐴𝐵) / (𝐶𝐷)) = ((𝐵𝐴) / (𝐷𝐶)))
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