HomeTheorem List (p. 87 of 140)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | diveqap1 8601 | Equality in terms of unit ratio. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵)) | ||
| Theorem | divnegap 8602 | Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) | ||
| Theorem | muldivdirap 8603 | 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 8604 | 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 8605 | 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 8606 | 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 8607 | 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 8608 | 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 8609 | 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 8610 | 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 8611 | 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 8612 | 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 8613 | 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 8614 | 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 8615 | 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 8616 | 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 8617 | 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 8618 | 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 8619 | 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 8620 | 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 8621 | 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 8622 | 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 8623 | 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 8624 | 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 8625 | 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 8626 | Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ) | ||
| Theorem | redivclap 8627 | Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℝ) | ||
| Theorem | eqneg 8628 | 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 8629 | A complex number equals its negative iff it is zero. Deduction form of eqneg 8628. (Contributed by David Moews, 28-Feb-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 = -𝐴 ↔ 𝐴 = 0)) | ||
| Theorem | eqnegad 8630 | If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 8628. (Contributed by David Moews, 28-Feb-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = -𝐴) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
| Theorem | div2negap 8631 | Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵)) | ||
| Theorem | divneg2ap 8632 | 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 8633 | Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 # 0 → (1 / 𝐴) ∈ ℂ) | ||
| Theorem | recap0apzi 8634 | 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 8635 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 # 0 → (𝐴 · (1 / 𝐴)) = 1) | ||
| Theorem | div1i 8636 | A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 / 1) = 𝐴 | ||
| Theorem | eqnegi 8637 | A number equal to its negative is zero. (Contributed by NM, 29-May-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 = -𝐴 ↔ 𝐴 = 0) | ||
| Theorem | recclapi 8638 | Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0 ⇒ ⊢ (1 / 𝐴) ∈ ℂ | ||
| Theorem | recidapi 8639 | Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0 ⇒ ⊢ (𝐴 · (1 / 𝐴)) = 1 | ||
| Theorem | recrecapi 8640 | 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 8641 | A number divided by itself is one. (Contributed by NM, 9-Feb-1995.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0 ⇒ ⊢ (𝐴 / 𝐴) = 1 | ||
| Theorem | div0api 8642 | Division into zero is zero. (Contributed by NM, 12-Aug-1999.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0 ⇒ ⊢ (0 / 𝐴) = 0 | ||
| Theorem | divclapzi 8643 | Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℂ) | ||
| Theorem | divcanap1zi 8644 | A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) | ||
| Theorem | divcanap2zi 8645 | A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴) | ||
| Theorem | divrecapzi 8646 | 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 8647 | A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴) | ||
| Theorem | divcanap4zi 8648 | A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) | ||
| Theorem | rec11api 8649 | 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 8650 | Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0 ⇒ ⊢ (𝐴 / 𝐵) ∈ ℂ | ||
| Theorem | divcanap2i 8651 | A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0 ⇒ ⊢ (𝐵 · (𝐴 / 𝐵)) = 𝐴 | ||
| Theorem | divcanap1i 8652 | A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0 ⇒ ⊢ ((𝐴 / 𝐵) · 𝐵) = 𝐴 | ||
| Theorem | divrecapi 8653 | 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 8654 | A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0 ⇒ ⊢ ((𝐵 · 𝐴) / 𝐵) = 𝐴 | ||
| Theorem | divcanap4i 8655 | A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 # 0 ⇒ ⊢ ((𝐴 · 𝐵) / 𝐵) = 𝐴 | ||
| Theorem | divap0i 8656 | 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 8657 | 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 8658 | 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 8659 | 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 8660 | 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 8661 | 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 8662 | 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 8663 | 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 8664 | 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 8665 | 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 8666 | 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 8667 | 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 8668 | 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 8669 | 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 8670 | 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 8671 | 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 8672 | Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐴 # 0 → (1 / 𝐴) ∈ ℝ) | ||
| Theorem | rerecclapi 8673 | Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 # 0 ⇒ ⊢ (1 / 𝐴) ∈ ℝ | ||
| Theorem | redivclapzi 8674 | Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℝ) | ||
| Theorem | redivclapi 8675 | Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐵 # 0 ⇒ ⊢ (𝐴 / 𝐵) ∈ ℝ | ||
| Theorem | div1d 8676 | A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 / 1) = 𝐴) | ||
| Theorem | recclapd 8677 | Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) | ||
| Theorem | recap0d 8678 | 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 8679 | 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 8680 | 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 8681 | 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 8682 | 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 8683 | Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → (0 / 𝐴) = 0) | ||
| Theorem | apmul1 8684 | 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 8685 | 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 8686 | 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 8687 | 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 8688 | 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 8689 | 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 8690 | 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 8691 | 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 8692 | 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 8693 | 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 8694 | 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 8695 | The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 8601. Generalization of diveqap1d 8694. (Contributed by Jim Kingdon, 19-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵)) | ||
| Theorem | diveqap0ad 8696 | A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 8578. (Contributed by Jim Kingdon, 19-Mar-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0)) | ||
| Theorem | divap1d 8697 | 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 8698 | 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 8699 | 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 8700 | 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)) | ||
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