HomeTheorem List (p. 91 of 171)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | muldivdirap 9001 | 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))) | ||
| Theorem | divsubdirap 9002 | 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 | recrecap 9003 | 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) | ||
| Theorem | rec11ap 9004 | 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)) | ||
| Theorem | rec11rap 9005 | Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) = 𝐵 ↔ (1 / 𝐵) = 𝐴)) | ||
| Theorem | divmuldivap 9006 | 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))) | ||
| Theorem | divdivdivap 9007 | 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))) | ||
| Theorem | divcanap5 9008 | 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)) | ||
| Theorem | divmul13ap 9009 | 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))) | ||
| Theorem | divmul24ap 9010 | 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))) | ||
| Theorem | divmuleqap 9011 | 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))) | ||
| Theorem | recdivap 9012 | 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)) | ||
| Theorem | divcanap6 9013 | Cancellation of inverted fractions. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1) | ||
| Theorem | divdiv32ap 9014 | 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)) | ||
| Theorem | divcanap7 9015 | 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)) | ||
| Theorem | dmdcanap 9016 | 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)) | ||
| Theorem | divdivap1 9017 | 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))) | ||
| Theorem | divdivap2 9018 | 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)) | ||
| Theorem | recdivap2 9019 | 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))) | ||
| Theorem | ddcanap 9020 | Cancellation in a double division. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 / (𝐴 / 𝐵)) = 𝐵) | ||
| Theorem | divadddivap 9021 | 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))) | ||
| Theorem | divsubdivap 9022 | 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))) | ||
| Theorem | conjmulap 9023 | 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)) | ||
| Theorem | rerecclap 9024 | Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ) | ||
| Theorem | redivclap 9025 | Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℝ) | ||
| Theorem | eqneg 9026 | 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 9027 | A complex number equals its negative iff it is zero. Deduction form of eqneg 9026. (Contributed by David Moews, 28-Feb-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 = -𝐴 ↔ 𝐴 = 0)) | ||
| Theorem | eqnegad 9028 | If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 9026. (Contributed by David Moews, 28-Feb-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = -𝐴) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
| Theorem | div2negap 9029 | Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵)) | ||
| Theorem | divneg2ap 9030 | Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐴 / 𝐵) = (𝐴 / -𝐵)) | ||
| Theorem | recclapzi 9031 | Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 # 0 → (1 / 𝐴) ∈ ℂ) | ||
| Theorem | recap0apzi 9032 | 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) | ||
| Theorem | recidapzi 9033 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 # 0 → (𝐴 · (1 / 𝐴)) = 1) | ||
| Theorem | div1i 9034 | A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 / 1) = 𝐴 | ||
| Theorem | eqnegi 9035 | A number equal to its negative is zero. (Contributed by NM, 29-May-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 = -𝐴 ↔ 𝐴 = 0) | ||
| Theorem | recclapi 9036 | Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0 ⇒ ⊢ (1 / 𝐴) ∈ ℂ | ||
| Theorem | recidapi 9037 | Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0 ⇒ ⊢ (𝐴 · (1 / 𝐴)) = 1 | ||
| Theorem | recrecapi 9038 | 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 | dividapi 9039 | A number divided by itself is one. (Contributed by NM, 9-Feb-1995.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0 ⇒ ⊢ (𝐴 / 𝐴) = 1 | ||
| Theorem | div0api 9040 | Division into zero is zero. (Contributed by NM, 12-Aug-1999.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0 ⇒ ⊢ (0 / 𝐴) = 0 | ||
| Theorem | divclapzi 9041 | Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℂ) | ||
| Theorem | divcanap1zi 9042 | A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) | ||
| Theorem | divcanap2zi 9043 | A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴) | ||
| Theorem | divrecapzi 9044 | Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) | ||
| Theorem | divcanap3zi 9045 | A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴) | ||
| Theorem | divcanap4zi 9046 | A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) | ||
| Theorem | rec11api 9047 | 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)) | ||
| Theorem | divclapi 9048 | Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0 ⇒ ⊢ (𝐴 / 𝐵) ∈ ℂ | ||
| Theorem | divcanap2i 9049 | A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0 ⇒ ⊢ (𝐵 · (𝐴 / 𝐵)) = 𝐴 | ||
| Theorem | divcanap1i 9050 | A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0 ⇒ ⊢ ((𝐴 / 𝐵) · 𝐵) = 𝐴 | ||
| Theorem | divrecapi 9051 | Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0 ⇒ ⊢ (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)) | ||
| Theorem | divcanap3i 9052 | A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0 ⇒ ⊢ ((𝐵 · 𝐴) / 𝐵) = 𝐴 | ||
| Theorem | divcanap4i 9053 | A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0 ⇒ ⊢ ((𝐴 · 𝐵) / 𝐵) = 𝐴 | ||
| Theorem | divap0i 9054 | 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 | ||
| Theorem | rec11apii 9055 | 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) | ||
| Theorem | divassapzi 9056 | 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))) | ||
| Theorem | divmulapzi 9057 | 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)) | ||
| Theorem | divdirapzi 9058 | 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))) | ||
| Theorem | divdiv23apzi 9059 | 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)) | ||
| Theorem | divmulapi 9060 | 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) | ||
| Theorem | divdiv32api 9061 | 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) | ||
| Theorem | divassapi 9062 | 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)) | ||
| Theorem | divdirapi 9063 | 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)) | ||
| Theorem | div23api 9064 | 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) | ||
| Theorem | div11api 9065 | 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) | ||
| Theorem | divmuldivapi 9066 | 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)) | ||
| Theorem | divmul13api 9067 | 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)) | ||
| Theorem | divadddivapi 9068 | 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)) | ||
| Theorem | divdivdivapi 9069 | 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)) | ||
| Theorem | rerecclapzi 9070 | Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐴 # 0 → (1 / 𝐴) ∈ ℝ) | ||
| Theorem | rerecclapi 9071 | Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 # 0 ⇒ ⊢ (1 / 𝐴) ∈ ℝ | ||
| Theorem | redivclapzi 9072 | Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℝ) | ||
| Theorem | redivclapi 9073 | Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐵 # 0 ⇒ ⊢ (𝐴 / 𝐵) ∈ ℝ | ||
| Theorem | div1d 9074 | A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 / 1) = 𝐴) | ||
| Theorem | recclapd 9075 | Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) | ||
| Theorem | recap0d 9076 | The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (1 / 𝐴) # 0) | ||
| Theorem | recidapd 9077 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (𝐴 · (1 / 𝐴)) = 1) | ||
| Theorem | recidap2d 9078 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → ((1 / 𝐴) · 𝐴) = 1) | ||
| Theorem | recrecapd 9079 | A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (1 / (1 / 𝐴)) = 𝐴) | ||
| Theorem | dividapd 9080 | A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐴) = 1) | ||
| Theorem | div0apd 9081 | Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (0 / 𝐴) = 0) | ||
| Theorem | apmul1 9082 | 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))) | ||
| Theorem | apmul2 9083 | 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))) | ||
| Theorem | divclapd 9084 | Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℂ) | ||
| Theorem | divcanap1d 9085 | A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) | ||
| Theorem | divcanap2d 9086 | A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐵 · (𝐴 / 𝐵)) = 𝐴) | ||
| Theorem | divrecapd 9087 | Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) | ||
| Theorem | divrecap2d 9088 | Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) | ||
| Theorem | divcanap3d 9089 | A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐵 · 𝐴) / 𝐵) = 𝐴) | ||
| Theorem | divcanap4d 9090 | A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) | ||
| Theorem | diveqap0d 9091 | If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → (𝐴 / 𝐵) = 0) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
| Theorem | diveqap1d 9092 | Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → (𝐴 / 𝐵) = 1) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
| Theorem | diveqap1ad 9093 | The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 8999. Generalization of diveqap1d 9092. (Contributed by Jim Kingdon, 19-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵)) | ||
| Theorem | diveqap0ad 9094 | A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 8976. (Contributed by Jim Kingdon, 19-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0)) | ||
| Theorem | divap1d 9095 | If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝐴 # 𝐵) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) # 1) | ||
| Theorem | divap0bd 9096 | A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 # 0 ↔ (𝐴 / 𝐵) # 0)) | ||
| Theorem | divnegapd 9097 | Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) | ||
| Theorem | divneg2apd 9098 | Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → -(𝐴 / 𝐵) = (𝐴 / -𝐵)) | ||
| Theorem | div2negapd 9099 | Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (-𝐴 / -𝐵) = (𝐴 / 𝐵)) | ||
| Theorem | divap0d 9100 | The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) # 0) | ||
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