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Theorem List for Intuitionistic Logic Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdivmulassap 8101 An associative law for division and multiplication. (Contributed by Jim Kingdon, 24-Jan-2022.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = ((𝐴 · 𝐵) · (𝐶 / 𝐷)))

Theoremdivmulasscomap 8102 An associative/commutative law for division and multiplication. (Contributed by Jim Kingdon, 24-Jan-2022.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = (𝐵 · ((𝐴 · 𝐶) / 𝐷)))

Theoremdivdirap 8103 Distribution of division over addition. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))

Theoremdivcanap3 8104 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴)

Theoremdivcanap4 8105 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴)

Theoremdiv11ap 8106 One-to-one relationship for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵))

Theoremdividap 8107 A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 / 𝐴) = 1)

Theoremdiv0ap 8108 Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (0 / 𝐴) = 0)

Theoremdiv1 8109 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 8110 1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
(1 / 1) = 1

Theoremdiveqap1 8111 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵))

Theoremdivnegap 8112 Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵))

Theoremmuldivdirap 8113 Distribution of division over addition with a multiplication. (Contributed by Jim Kingdon, 11-Nov-2021.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (((𝐶 · 𝐴) + 𝐵) / 𝐶) = (𝐴 + (𝐵 / 𝐶)))

Theoremdivsubdirap 8114 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)))

Theoremrecrecap 8115 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / (1 / 𝐴)) = 𝐴)

Theoremrec11ap 8116 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵))

Theoremrec11rap 8117 Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) = 𝐵 ↔ (1 / 𝐵) = 𝐴))

Theoremdivmuldivap 8118 Multiplication of two ratios. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐴 · 𝐵) / (𝐶 · 𝐷)))

Theoremdivdivdivap 8119 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶)))

Theoremdivcanap5 8120 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵))

Theoremdivmul13ap 8121 Swap the denominators in the product 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)) = ((𝐵 / 𝐶) · (𝐴 / 𝐷)))

Theoremdivmul24ap 8122 Swap the numerators in the product 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)))

Theoremdivmuleqap 8123 Cross-multiply in an equality of 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)))

Theoremrecdivap 8124 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 8125 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 8126 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 8127 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 8128 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 8129 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 8130 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 8131 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 8132 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 8133 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 8134 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 8135 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 8136 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ)

Theoremredivclap 8137 Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℝ)

Theoremeqneg 8138 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 8139 A complex number equals its negative iff it is zero. Deduction form of eqneg 8138. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 = -𝐴𝐴 = 0))

Theoremeqnegad 8140 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 8138. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 = -𝐴)       (𝜑𝐴 = 0)

Theoremdiv2negap 8141 Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵))

Theoremdivneg2ap 8142 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 8143 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ       (𝐴 # 0 → (1 / 𝐴) ∈ ℂ)

Theoremrecap0apzi 8144 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 8145 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ       (𝐴 # 0 → (𝐴 · (1 / 𝐴)) = 1)

Theoremdiv1i 8146 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
𝐴 ∈ ℂ       (𝐴 / 1) = 𝐴

Theoremeqnegi 8147 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 = -𝐴𝐴 = 0)

Theoremrecclapi 8148 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
𝐴 ∈ ℂ    &   𝐴 # 0       (1 / 𝐴) ∈ ℂ

Theoremrecidapi 8149 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐴 # 0       (𝐴 · (1 / 𝐴)) = 1

Theoremrecrecapi 8150 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 8151 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐴 # 0       (𝐴 / 𝐴) = 1

Theoremdiv0api 8152 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
𝐴 ∈ ℂ    &   𝐴 # 0       (0 / 𝐴) = 0

Theoremdivclapzi 8153 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℂ)

Theoremdivcanap1zi 8154 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)

Theoremdivcanap2zi 8155 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)

Theoremdivrecapzi 8156 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))

Theoremdivcanap3zi 8157 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)

Theoremdivcanap4zi 8158 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)

Theoremrec11api 8159 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 8160 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐴 / 𝐵) ∈ ℂ

Theoremdivcanap2i 8161 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐵 · (𝐴 / 𝐵)) = 𝐴

Theoremdivcanap1i 8162 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐴 / 𝐵) · 𝐵) = 𝐴

Theoremdivrecapi 8163 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))

Theoremdivcanap3i 8164 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐵 · 𝐴) / 𝐵) = 𝐴

Theoremdivcanap4i 8165 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐴 · 𝐵) / 𝐵) = 𝐴

Theoremdivap0i 8166 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 8167 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 8168 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 8169 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 8170 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 8171 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 8172 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 8173 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 8174 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 8175 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 8176 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 8177 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 8178 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 8179 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 8180 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 8181 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 8182 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ       (𝐴 # 0 → (1 / 𝐴) ∈ ℝ)

Theoremrerecclapi 8183 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐴 # 0       (1 / 𝐴) ∈ ℝ

Theoremredivclapzi 8184 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℝ)

Theoremredivclapi 8185 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐵 # 0       (𝐴 / 𝐵) ∈ ℝ

Theoremdiv1d 8186 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 / 1) = 𝐴)

Theoremrecclapd 8187 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (1 / 𝐴) ∈ ℂ)

Theoremrecap0d 8188 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 8189 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (𝐴 · (1 / 𝐴)) = 1)

Theoremrecidap2d 8190 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → ((1 / 𝐴) · 𝐴) = 1)

Theoremrecrecapd 8191 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 8192 A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (𝐴 / 𝐴) = 1)

Theoremdiv0apd 8193 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (0 / 𝐴) = 0)

Theoremapmul1 8194 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)))

Theoremdivclapd 8195 Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / 𝐵) ∈ ℂ)

Theoremdivcanap1d 8196 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)

Theoremdivcanap2d 8197 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)

Theoremdivrecapd 8198 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 8199 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 8200 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)

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