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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | recrec 10601 | A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 26-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / (1 / 𝐴)) = 𝐴) | ||
Theorem | rec11 10602 | Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | rec11r 10603 | Mutual reciprocals. (Contributed by Paul Chapman, 18-Oct-2007.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((1 / 𝐴) = 𝐵 ↔ (1 / 𝐵) = 𝐴)) | ||
Theorem | divmuldiv 10604 | Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by NM, 1-Aug-2004.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐴 · 𝐵) / (𝐶 · 𝐷))) | ||
Theorem | divdivdiv 10605 | Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by NM, 2-Aug-2004.) |
⊢ (((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶))) | ||
Theorem | divcan5 10606 | Cancellation of common factor in a ratio. (Contributed by NM, 9-Jan-2006.) |
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵)) | ||
Theorem | divmul13 10607 | Swap the denominators in the product of two ratios. (Contributed by NM, 3-May-2005.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐵 / 𝐶) · (𝐴 / 𝐷))) | ||
Theorem | divmul24 10608 | Swap the numerators in the product of two ratios. (Contributed by NM, 3-May-2005.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐴 / 𝐷) · (𝐵 / 𝐶))) | ||
Theorem | divmuleq 10609 | Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 23-Feb-2014.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐶) = (𝐵 / 𝐷) ↔ (𝐴 · 𝐷) = (𝐵 · 𝐶))) | ||
Theorem | recdiv 10610 | The reciprocal of a ratio. (Contributed by NM, 3-Aug-2004.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴)) | ||
Theorem | divcan6 10611 | Cancellation of inverted fractions. (Contributed by NM, 28-Dec-2007.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1) | ||
Theorem | divdiv32 10612 | Swap denominators in a division. (Contributed by NM, 2-Aug-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵)) | ||
Theorem | divcan7 10613 | Cancel equal divisors in a division. (Contributed by Jeff Hankins, 29-Sep-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) / (𝐵 / 𝐶)) = (𝐴 / 𝐵)) | ||
Theorem | dmdcan 10614 | Cancellation law for division and multiplication. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by Fan Zheng, 3-Jul-2016.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · (𝐶 / 𝐴)) = (𝐶 / 𝐵)) | ||
Theorem | divdiv1 10615 | Division into a fraction. (Contributed by NM, 31-Dec-2007.) |
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶))) | ||
Theorem | divdiv2 10616 | Division by a fraction. (Contributed by NM, 27-Dec-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 / (𝐵 / 𝐶)) = ((𝐴 · 𝐶) / 𝐵)) | ||
Theorem | recdiv2 10617 | Division into a reciprocal. (Contributed by NM, 19-Oct-2007.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵))) | ||
Theorem | ddcan 10618 | Cancellation in a double division. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 / (𝐴 / 𝐵)) = 𝐵) | ||
Theorem | divadddiv 10619 | Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 2-May-2016.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐶) + (𝐵 / 𝐷)) = (((𝐴 · 𝐷) + (𝐵 · 𝐶)) / (𝐶 · 𝐷))) | ||
Theorem | divsubdiv 10620 | Subtraction of two ratios. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 2-May-2016.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))) → ((𝐴 / 𝐶) − (𝐵 / 𝐷)) = (((𝐴 · 𝐷) − (𝐵 · 𝐶)) / (𝐶 · 𝐷))) | ||
Theorem | conjmul 10621 | Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 12-Nov-2006.) |
⊢ (((𝑃 ∈ ℂ ∧ 𝑃 ≠ 0) ∧ (𝑄 ∈ ℂ ∧ 𝑄 ≠ 0)) → (((1 / 𝑃) + (1 / 𝑄)) = 1 ↔ ((𝑃 − 1) · (𝑄 − 1)) = 1)) | ||
Theorem | rereccl 10622 | Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℝ) | ||
Theorem | redivcl 10623 | Closure law for division of reals. (Contributed by NM, 27-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℝ) | ||
Theorem | eqneg 10624 | 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 10625 | A complex number equals its negative iff it is zero. Deduction form of eqneg 10624. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 = -𝐴 ↔ 𝐴 = 0)) | ||
Theorem | eqnegad 10626 | If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 10624. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = -𝐴) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
Theorem | div2neg 10627 | Quotient of two negatives. (Contributed by Paul Chapman, 10-Nov-2012.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵)) | ||
Theorem | divneg2 10628 | Move negative sign inside of a division. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → -(𝐴 / 𝐵) = (𝐴 / -𝐵)) | ||
Theorem | recclzi 10629 | Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 ≠ 0 → (1 / 𝐴) ∈ ℂ) | ||
Theorem | recne0zi 10630 | The reciprocal of a nonzero number is nonzero. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 ≠ 0 → (1 / 𝐴) ≠ 0) | ||
Theorem | recidzi 10631 | Multiplication of a number and its reciprocal. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 ≠ 0 → (𝐴 · (1 / 𝐴)) = 1) | ||
Theorem | div1i 10632 | A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 / 1) = 𝐴 | ||
Theorem | eqnegi 10633 | A number equal to its negative is zero. (Contributed by NM, 29-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 = -𝐴 ↔ 𝐴 = 0) | ||
Theorem | reccli 10634 | Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠ 0 ⇒ ⊢ (1 / 𝐴) ∈ ℂ | ||
Theorem | recidi 10635 | Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠ 0 ⇒ ⊢ (𝐴 · (1 / 𝐴)) = 1 | ||
Theorem | recreci 10636 | 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 | dividi 10637 | A number divided by itself is one. (Contributed by NM, 9-Feb-1995.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠ 0 ⇒ ⊢ (𝐴 / 𝐴) = 1 | ||
Theorem | div0i 10638 | Division into zero is zero. (Contributed by NM, 12-Aug-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠ 0 ⇒ ⊢ (0 / 𝐴) = 0 | ||
Theorem | divclzi 10639 | 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) | ||
Theorem | divcan1zi 10640 | A cancellation law for division. (Contributed by NM, 2-Oct-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) | ||
Theorem | divcan2zi 10641 | A cancellation law for division. (Contributed by NM, 10-Aug-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴) | ||
Theorem | divreczi 10642 | 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))) | ||
Theorem | divcan3zi 10643 | A cancellation law for division. (Eliminates a hypothesis of divcan3i 10650 with the weak deduction theorem.) (Contributed by NM, 3-Feb-2004.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴) | ||
Theorem | divcan4zi 10644 | A cancellation law for division. (Contributed by NM, 12-Oct-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) | ||
Theorem | rec11i 10645 | 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)) | ||
Theorem | divcli 10646 | 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 | ||
Theorem | divcan2i 10647 | A cancellation law for division. (Contributed by NM, 9-Feb-1995.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 ≠ 0 ⇒ ⊢ (𝐵 · (𝐴 / 𝐵)) = 𝐴 | ||
Theorem | divcan1i 10648 | A cancellation law for division. (Contributed by NM, 18-May-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 ≠ 0 ⇒ ⊢ ((𝐴 / 𝐵) · 𝐵) = 𝐴 | ||
Theorem | divreci 10649 | 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)) | ||
Theorem | divcan3i 10650 | A cancellation law for division. (Contributed by NM, 16-Feb-1995.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 ≠ 0 ⇒ ⊢ ((𝐵 · 𝐴) / 𝐵) = 𝐴 | ||
Theorem | divcan4i 10651 | A cancellation law for division. (Contributed by NM, 18-May-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 ≠ 0 ⇒ ⊢ ((𝐴 · 𝐵) / 𝐵) = 𝐴 | ||
Theorem | divne0i 10652 | The ratio of nonzero numbers is nonzero. (Contributed by NM, 9-Feb-1995.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐴 ≠ 0 & ⊢ 𝐵 ≠ 0 ⇒ ⊢ (𝐴 / 𝐵) ≠ 0 | ||
Theorem | rec11ii 10653 | 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) | ||
Theorem | divasszi 10654 | 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))) | ||
Theorem | divmulzi 10655 | 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)) | ||
Theorem | divdirzi 10656 | 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))) | ||
Theorem | divdiv23zi 10657 | 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)) | ||
Theorem | divmuli 10658 | 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) | ||
Theorem | divdiv32i 10659 | 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) | ||
Theorem | divassi 10660 | 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)) | ||
Theorem | divdiri 10661 | 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)) | ||
Theorem | div23i 10662 | 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) | ||
Theorem | div11i 10663 | 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) | ||
Theorem | divmuldivi 10664 | 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)) | ||
Theorem | divmul13i 10665 | 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)) | ||
Theorem | divadddivi 10666 | 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)) | ||
Theorem | divdivdivi 10667 | 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)) | ||
Theorem | rerecclzi 10668 | Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐴 ≠ 0 → (1 / 𝐴) ∈ ℝ) | ||
Theorem | rereccli 10669 | Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 ≠ 0 ⇒ ⊢ (1 / 𝐴) ∈ ℝ | ||
Theorem | redivclzi 10670 | Closure law for division of reals. (Contributed by NM, 9-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐵 ≠ 0 → (𝐴 / 𝐵) ∈ ℝ) | ||
Theorem | redivcli 10671 | Closure law for division of reals. (Contributed by NM, 9-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐵 ≠ 0 ⇒ ⊢ (𝐴 / 𝐵) ∈ ℝ | ||
Theorem | div1d 10672 | A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 / 1) = 𝐴) | ||
Theorem | reccld 10673 | Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) | ||
Theorem | recne0d 10674 | The reciprocal of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (1 / 𝐴) ≠ 0) | ||
Theorem | recidd 10675 | Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 · (1 / 𝐴)) = 1) | ||
Theorem | recid2d 10676 | Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → ((1 / 𝐴) · 𝐴) = 1) | ||
Theorem | recrecd 10677 | A number is equal to the reciprocal of its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (1 / (1 / 𝐴)) = 𝐴) | ||
Theorem | dividd 10678 | A number divided by itself is one. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐴) = 1) | ||
Theorem | div0d 10679 | Division into zero is zero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (0 / 𝐴) = 0) | ||
Theorem | divcld 10680 | Closure law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℂ) | ||
Theorem | divcan1d 10681 | A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) | ||
Theorem | divcan2d 10682 | A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (𝐵 · (𝐴 / 𝐵)) = 𝐴) | ||
Theorem | divrecd 10683 | Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) | ||
Theorem | divrec2d 10684 | Relationship between division and reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) | ||
Theorem | divcan3d 10685 | A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐵 · 𝐴) / 𝐵) = 𝐴) | ||
Theorem | divcan4d 10686 | A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) | ||
Theorem | diveq0d 10687 | A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) & ⊢ (𝜑 → (𝐴 / 𝐵) = 0) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
Theorem | diveq1d 10688 | Equality in terms of unit ratio. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) & ⊢ (𝜑 → (𝐴 / 𝐵) = 1) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | diveq1ad 10689 | The quotient of two complex numbers is one iff they are equal. Deduction form of diveq1 10597. Generalization of diveq1d 10688. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵)) | ||
Theorem | diveq0ad 10690 | A fraction of complex numbers is zero iff its numerator is. Deduction form of diveq0 10574. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0)) | ||
Theorem | divne1d 10691 | If two complex numbers are unequal, their quotient is not one. Contrapositive of diveq1d 10688. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) ≠ 1) | ||
Theorem | divne0bd 10692 | A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 ≠ 0 ↔ (𝐴 / 𝐵) ≠ 0)) | ||
Theorem | divnegd 10693 | Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) | ||
Theorem | divneg2d 10694 | Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → -(𝐴 / 𝐵) = (𝐴 / -𝐵)) | ||
Theorem | div2negd 10695 | Quotient of two negatives. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (-𝐴 / -𝐵) = (𝐴 / 𝐵)) | ||
Theorem | divne0d 10696 | The ratio of nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 / 𝐵) ≠ 0) | ||
Theorem | recdivd 10697 | The reciprocal of a ratio. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴)) | ||
Theorem | recdiv2d 10698 | Division into a reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵))) | ||
Theorem | divcan6d 10699 | Cancellation of inverted fractions. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1) | ||
Theorem | ddcand 10700 | Cancellation in a double division. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 / (𝐴 / 𝐵)) = 𝐵) |
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