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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | recid2 11301 | Multiplication of a number and its reciprocal. (Contributed by NM, 22-Jun-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((1 / 𝐴) · 𝐴) = 1) | ||
Theorem | divrec 11302 | Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) | ||
Theorem | divrec2 11303 | Relationship between division and reciprocal. (Contributed by NM, 7-Feb-2006.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) | ||
Theorem | divass 11304 | An associative law for division. (Contributed by NM, 2-Aug-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))) | ||
Theorem | div23 11305 | A commutative/associative law for division. (Contributed by NM, 2-Aug-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)) | ||
Theorem | div32 11306 | A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = (𝐴 · (𝐶 / 𝐵))) | ||
Theorem | div13 11307 | A commutative/associative law for division. (Contributed by NM, 22-Apr-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = ((𝐶 / 𝐵) · 𝐴)) | ||
Theorem | div12 11308 | A commutative/associative law for division. (Contributed by NM, 30-Apr-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶))) | ||
Theorem | divmulass 11309 | An associative law for division and multiplication. (Contributed by AV, 10-Jul-2021.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = ((𝐴 · 𝐵) · (𝐶 / 𝐷))) | ||
Theorem | divmulasscom 11310 | An associative/commutative law for division and multiplication. (Contributed by AV, 10-Jul-2021.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 ≠ 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = (𝐵 · ((𝐴 · 𝐶) / 𝐷))) | ||
Theorem | divdir 11311 | Distribution of division over addition. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶))) | ||
Theorem | divcan3 11312 | A cancellation law for division. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴) | ||
Theorem | divcan4 11313 | A cancellation law for division. (Contributed by NM, 8-Feb-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴) | ||
Theorem | div11 11314 | One-to-one relationship for division. (Contributed by NM, 20-Apr-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | divid 11315 | A number divided by itself is one. (Contributed by NM, 1-Aug-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴 / 𝐴) = 1) | ||
Theorem | div0 11316 | Division into zero is zero. (Contributed by NM, 14-Mar-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0 / 𝐴) = 0) | ||
Theorem | div1 11317 | 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) | ||
Theorem | 1div1e1 11318 | 1 divided by 1 is 1. (Contributed by David A. Wheeler, 7-Dec-2018.) |
⊢ (1 / 1) = 1 | ||
Theorem | diveq1 11319 | Equality in terms of unit ratio. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵)) | ||
Theorem | divneg 11320 | Move negative sign inside of a division. (Contributed by NM, 17-Sep-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) | ||
Theorem | muldivdir 11321 | Distribution of division over addition with a multiplication. (Contributed by AV, 1-Jul-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((𝐶 · 𝐴) + 𝐵) / 𝐶) = (𝐴 + (𝐵 / 𝐶))) | ||
Theorem | divsubdir 11322 | 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 | subdivcomb1 11323 | Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((𝐶 · 𝐴) − 𝐵) / 𝐶) = (𝐴 − (𝐵 / 𝐶))) | ||
Theorem | subdivcomb2 11324 | Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 − (𝐶 · 𝐵)) / 𝐶) = ((𝐴 / 𝐶) − 𝐵)) | ||
Theorem | recrec 11325 | 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 11326 | 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 11327 | 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 11328 | 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 11329 | 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 11330 | 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 11331 | 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 11332 | 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 11333 | 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 11334 | 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 11335 | 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 11336 | 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 11337 | 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 11338 | 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 11339 | 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 11340 | 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 11341 | 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 11342 | 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 11343 | 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 11344 | 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 11345 | 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 11346 | 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 11347 | 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 11348 | 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 11349 | A complex number equals its negative iff it is zero. Deduction form of eqneg 11348. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 = -𝐴 ↔ 𝐴 = 0)) | ||
Theorem | eqnegad 11350 | If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 11348. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = -𝐴) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
Theorem | div2neg 11351 | Quotient of two negatives. (Contributed by Paul Chapman, 10-Nov-2012.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵)) | ||
Theorem | divneg2 11352 | 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 11353 | Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 ≠ 0 → (1 / 𝐴) ∈ ℂ) | ||
Theorem | recne0zi 11354 | The reciprocal of a nonzero number is nonzero. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 ≠ 0 → (1 / 𝐴) ≠ 0) | ||
Theorem | recidzi 11355 | Multiplication of a number and its reciprocal. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 ≠ 0 → (𝐴 · (1 / 𝐴)) = 1) | ||
Theorem | div1i 11356 | A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 / 1) = 𝐴 | ||
Theorem | eqnegi 11357 | A number equal to its negative is zero. (Contributed by NM, 29-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 = -𝐴 ↔ 𝐴 = 0) | ||
Theorem | reccli 11358 | Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠ 0 ⇒ ⊢ (1 / 𝐴) ∈ ℂ | ||
Theorem | recidi 11359 | Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠ 0 ⇒ ⊢ (𝐴 · (1 / 𝐴)) = 1 | ||
Theorem | recreci 11360 | 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 11361 | A number divided by itself is one. (Contributed by NM, 9-Feb-1995.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠ 0 ⇒ ⊢ (𝐴 / 𝐴) = 1 | ||
Theorem | div0i 11362 | Division into zero is zero. (Contributed by NM, 12-Aug-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠ 0 ⇒ ⊢ (0 / 𝐴) = 0 | ||
Theorem | divclzi 11363 | 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 11364 | A cancellation law for division. (Contributed by NM, 2-Oct-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) | ||
Theorem | divcan2zi 11365 | A cancellation law for division. (Contributed by NM, 10-Aug-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴) | ||
Theorem | divreczi 11366 | 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 11367 | A cancellation law for division. (Eliminates a hypothesis of divcan3i 11374 with the weak deduction theorem.) (Contributed by NM, 3-Feb-2004.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴) | ||
Theorem | divcan4zi 11368 | A cancellation law for division. (Contributed by NM, 12-Oct-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴) | ||
Theorem | rec11i 11369 | 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 11370 | 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 11371 | A cancellation law for division. (Contributed by NM, 9-Feb-1995.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 ≠ 0 ⇒ ⊢ (𝐵 · (𝐴 / 𝐵)) = 𝐴 | ||
Theorem | divcan1i 11372 | A cancellation law for division. (Contributed by NM, 18-May-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 ≠ 0 ⇒ ⊢ ((𝐴 / 𝐵) · 𝐵) = 𝐴 | ||
Theorem | divreci 11373 | 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 11374 | A cancellation law for division. (Contributed by NM, 16-Feb-1995.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 ≠ 0 ⇒ ⊢ ((𝐵 · 𝐴) / 𝐵) = 𝐴 | ||
Theorem | divcan4i 11375 | A cancellation law for division. (Contributed by NM, 18-May-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐵 ≠ 0 ⇒ ⊢ ((𝐴 · 𝐵) / 𝐵) = 𝐴 | ||
Theorem | divne0i 11376 | 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 11377 | 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 11378 | 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 11379 | 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 11380 | 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 11381 | 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 11382 | 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 11383 | 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 11384 | 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 11385 | 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 11386 | 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 11387 | 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 11388 | 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 11389 | 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 11390 | 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 11391 | 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 11392 | Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐴 ≠ 0 → (1 / 𝐴) ∈ ℝ) | ||
Theorem | rereccli 11393 | Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 ≠ 0 ⇒ ⊢ (1 / 𝐴) ∈ ℝ | ||
Theorem | redivclzi 11394 | Closure law for division of reals. (Contributed by NM, 9-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐵 ≠ 0 → (𝐴 / 𝐵) ∈ ℝ) | ||
Theorem | redivcli 11395 | Closure law for division of reals. (Contributed by NM, 9-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐵 ≠ 0 ⇒ ⊢ (𝐴 / 𝐵) ∈ ℝ | ||
Theorem | div1d 11396 | A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 / 1) = 𝐴) | ||
Theorem | reccld 11397 | Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) | ||
Theorem | recne0d 11398 | 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 11399 | 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 11400 | 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) |
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