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Theorem List for Intuitionistic Logic Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrecdivap 8501 The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴))
 
Theoremdivcanap6 8502 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1)
 
Theoremdivdiv32ap 8503 Swap denominators in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵))
 
Theoremdivcanap7 8504 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) / (𝐵 / 𝐶)) = (𝐴 / 𝐵))
 
Theoremdmdcanap 8505 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · (𝐶 / 𝐴)) = (𝐶 / 𝐵))
 
Theoremdivdivap1 8506 Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶)))
 
Theoremdivdivap2 8507 Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 / (𝐵 / 𝐶)) = ((𝐴 · 𝐶) / 𝐵))
 
Theoremrecdivap2 8508 Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵)))
 
Theoremddcanap 8509 Cancellation in a double division. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 / (𝐴 / 𝐵)) = 𝐵)
 
Theoremdivadddivap 8510 Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) + (𝐵 / 𝐷)) = (((𝐴 · 𝐷) + (𝐵 · 𝐶)) / (𝐶 · 𝐷)))
 
Theoremdivsubdivap 8511 Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) − (𝐵 / 𝐷)) = (((𝐴 · 𝐷) − (𝐵 · 𝐶)) / (𝐶 · 𝐷)))
 
Theoremconjmulap 8512 Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝑃 ∈ ℂ ∧ 𝑃 # 0) ∧ (𝑄 ∈ ℂ ∧ 𝑄 # 0)) → (((1 / 𝑃) + (1 / 𝑄)) = 1 ↔ ((𝑃 − 1) · (𝑄 − 1)) = 1))
 
Theoremrerecclap 8513 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ)
 
Theoremredivclap 8514 Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℝ)
 
Theoremeqneg 8515 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 8516 A complex number equals its negative iff it is zero. Deduction form of eqneg 8515. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 = -𝐴𝐴 = 0))
 
Theoremeqnegad 8517 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 8515. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 = -𝐴)       (𝜑𝐴 = 0)
 
Theoremdiv2negap 8518 Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵))
 
Theoremdivneg2ap 8519 Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐴 / 𝐵) = (𝐴 / -𝐵))
 
Theoremrecclapzi 8520 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ       (𝐴 # 0 → (1 / 𝐴) ∈ ℂ)
 
Theoremrecap0apzi 8521 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ       (𝐴 # 0 → (1 / 𝐴) # 0)
 
Theoremrecidapzi 8522 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ       (𝐴 # 0 → (𝐴 · (1 / 𝐴)) = 1)
 
Theoremdiv1i 8523 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
𝐴 ∈ ℂ       (𝐴 / 1) = 𝐴
 
Theoremeqnegi 8524 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 = -𝐴𝐴 = 0)
 
Theoremrecclapi 8525 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
𝐴 ∈ ℂ    &   𝐴 # 0       (1 / 𝐴) ∈ ℂ
 
Theoremrecidapi 8526 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐴 # 0       (𝐴 · (1 / 𝐴)) = 1
 
Theoremrecrecapi 8527 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
 
Theoremdividapi 8528 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐴 # 0       (𝐴 / 𝐴) = 1
 
Theoremdiv0api 8529 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
𝐴 ∈ ℂ    &   𝐴 # 0       (0 / 𝐴) = 0
 
Theoremdivclapzi 8530 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℂ)
 
Theoremdivcanap1zi 8531 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
 
Theoremdivcanap2zi 8532 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
 
Theoremdivrecapzi 8533 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
 
Theoremdivcanap3zi 8534 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
 
Theoremdivcanap4zi 8535 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
Theoremrec11api 8536 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 # 0 ∧ 𝐵 # 0) → ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremdivclapi 8537 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐴 / 𝐵) ∈ ℂ
 
Theoremdivcanap2i 8538 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐵 · (𝐴 / 𝐵)) = 𝐴
 
Theoremdivcanap1i 8539 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐴 / 𝐵) · 𝐵) = 𝐴
 
Theoremdivrecapi 8540 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))
 
Theoremdivcanap3i 8541 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐵 · 𝐴) / 𝐵) = 𝐴
 
Theoremdivcanap4i 8542 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐴 · 𝐵) / 𝐵) = 𝐴
 
Theoremdivap0i 8543 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 # 0    &   𝐵 # 0       (𝐴 / 𝐵) # 0
 
Theoremrec11apii 8544 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 # 0    &   𝐵 # 0       ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵)
 
Theoremdivassapzi 8545 An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐶 # 0 → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)))
 
Theoremdivmulapzi 8546 Relationship between division and multiplication. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐵 # 0 → ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴))
 
Theoremdivdirapzi 8547 Distribution of division over addition. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐶 # 0 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))
 
Theoremdivdiv23apzi 8548 Swap denominators in a division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐵 # 0 ∧ 𝐶 # 0) → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵))
 
Theoremdivmulapi 8549 Relationship between division and multiplication. (Contributed by Jim Kingdon, 29-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐵 # 0       ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴)
 
Theoremdivdiv32api 8550 Swap denominators in a division. (Contributed by Jim Kingdon, 29-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐵 # 0    &   𝐶 # 0       ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵)
 
Theoremdivassapi 8551 An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0       ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))
 
Theoremdivdirapi 8552 Distribution of division over addition. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0       ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶))
 
Theoremdiv23api 8553 A commutative/associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0       ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)
 
Theoremdiv11api 8554 One-to-one relationship for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0       ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵)
 
Theoremdivmuldivapi 8555 Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 # 0    &   𝐷 # 0       ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷))
 
Theoremdivmul13api 8556 Swap denominators of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 # 0    &   𝐷 # 0       ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐶 / 𝐵) · (𝐴 / 𝐷))
 
Theoremdivadddivapi 8557 Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 # 0    &   𝐷 # 0       ((𝐴 / 𝐵) + (𝐶 / 𝐷)) = (((𝐴 · 𝐷) + (𝐶 · 𝐵)) / (𝐵 · 𝐷))
 
Theoremdivdivdivapi 8558 Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 # 0    &   𝐷 # 0    &   𝐶 # 0       ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶))
 
Theoremrerecclapzi 8559 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ       (𝐴 # 0 → (1 / 𝐴) ∈ ℝ)
 
Theoremrerecclapi 8560 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐴 # 0       (1 / 𝐴) ∈ ℝ
 
Theoremredivclapzi 8561 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℝ)
 
Theoremredivclapi 8562 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐵 # 0       (𝐴 / 𝐵) ∈ ℝ
 
Theoremdiv1d 8563 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 / 1) = 𝐴)
 
Theoremrecclapd 8564 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (1 / 𝐴) ∈ ℂ)
 
Theoremrecap0d 8565 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (1 / 𝐴) # 0)
 
Theoremrecidapd 8566 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (𝐴 · (1 / 𝐴)) = 1)
 
Theoremrecidap2d 8567 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → ((1 / 𝐴) · 𝐴) = 1)
 
Theoremrecrecapd 8568 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (1 / (1 / 𝐴)) = 𝐴)
 
Theoremdividapd 8569 A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (𝐴 / 𝐴) = 1)
 
Theoremdiv0apd 8570 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (0 / 𝐴) = 0)
 
Theoremapmul1 8571 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))
 
Theoremapmul2 8572 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐶 · 𝐴) # (𝐶 · 𝐵)))
 
Theoremdivclapd 8573 Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / 𝐵) ∈ ℂ)
 
Theoremdivcanap1d 8574 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
 
Theoremdivcanap2d 8575 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
 
Theoremdivrecapd 8576 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
 
Theoremdivrecap2d 8577 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴))
 
Theoremdivcanap3d 8578 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
 
Theoremdivcanap4d 8579 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
Theoremdiveqap0d 8580 If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑 → (𝐴 / 𝐵) = 0)       (𝜑𝐴 = 0)
 
Theoremdiveqap1d 8581 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑 → (𝐴 / 𝐵) = 1)       (𝜑𝐴 = 𝐵)
 
Theoremdiveqap1ad 8582 The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 8488. Generalization of diveqap1d 8581. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵))
 
Theoremdiveqap0ad 8583 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 8465. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0))
 
Theoremdivap1d 8584 If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐴 # 𝐵)       (𝜑 → (𝐴 / 𝐵) # 1)
 
Theoremdivap0bd 8585 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 # 0 ↔ (𝐴 / 𝐵) # 0))
 
Theoremdivnegapd 8586 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → -(𝐴 / 𝐵) = (-𝐴 / 𝐵))
 
Theoremdivneg2apd 8587 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → -(𝐴 / 𝐵) = (𝐴 / -𝐵))
 
Theoremdiv2negapd 8588 Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (-𝐴 / -𝐵) = (𝐴 / 𝐵))
 
Theoremdivap0d 8589 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / 𝐵) # 0)
 
Theoremrecdivapd 8590 The reciprocal of a ratio. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)       (𝜑 → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴))
 
Theoremrecdivap2d 8591 Division into a reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)       (𝜑 → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵)))
 
Theoremdivcanap6d 8592 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1)
 
Theoremddcanapd 8593 Cancellation in a double division. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / (𝐴 / 𝐵)) = 𝐵)
 
Theoremrec11apd 8594 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)    &   (𝜑 → (1 / 𝐴) = (1 / 𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremdivmulapd 8595 Relationship between division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴))
 
Theoremapdivmuld 8596 Relationship between division and multiplication. (Contributed by Jim Kingdon, 26-Dec-2022.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) # 𝐶 ↔ (𝐵 · 𝐶) # 𝐴))
 
Theoremdiv32apd 8597 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) · 𝐶) = (𝐴 · (𝐶 / 𝐵)))
 
Theoremdiv13apd 8598 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) · 𝐶) = ((𝐶 / 𝐵) · 𝐴))
 
Theoremdivdiv32apd 8599 Swap denominators in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵))
 
Theoremdivcanap5d 8600 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵))
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