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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | subsub4d 11601 | Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) − 𝐶) = (𝐴 − (𝐵 + 𝐶))) | ||
Theorem | sub32d 11602 | Swap the second and third terms in a double subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) − 𝐶) = ((𝐴 − 𝐶) − 𝐵)) | ||
Theorem | nnncand 11603 | Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − (𝐵 − 𝐶)) − 𝐶) = (𝐴 − 𝐵)) | ||
Theorem | nnncan1d 11604 | Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) − (𝐴 − 𝐶)) = (𝐶 − 𝐵)) | ||
Theorem | nnncan2d 11605 | Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐶) − (𝐵 − 𝐶)) = (𝐴 − 𝐵)) | ||
Theorem | npncan3d 11606 | Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) + (𝐶 − 𝐴)) = (𝐶 − 𝐵)) | ||
Theorem | pnpcand 11607 | Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵 − 𝐶)) | ||
Theorem | pnpcan2d 11608 | Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐶) − (𝐵 + 𝐶)) = (𝐴 − 𝐵)) | ||
Theorem | pnncand 11609 | Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − (𝐴 − 𝐶)) = (𝐵 + 𝐶)) | ||
Theorem | ppncand 11610 | Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 − 𝐵)) = (𝐴 + 𝐶)) | ||
Theorem | subcand 11611 | Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐴 − 𝐶)) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶) | ||
Theorem | subcan2d 11612 | Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐵 − 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | subcanad 11613 | Cancellation law for subtraction. Deduction form of subcan 11514. Generalization of subcand 11611. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐴 − 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | subneintrd 11614 | Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand 11611. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ≠ (𝐴 − 𝐶)) | ||
Theorem | subcan2ad 11615 | Cancellation law for subtraction. Deduction form of subcan2 11484. Generalization of subcan2d 11612. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | subneintr2d 11616 | Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 11612. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) ≠ (𝐵 − 𝐶)) | ||
Theorem | addsub4d 11617 | Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − (𝐶 + 𝐷)) = ((𝐴 − 𝐶) + (𝐵 − 𝐷))) | ||
Theorem | subadd4d 11618 | Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) − (𝐶 − 𝐷)) = ((𝐴 + 𝐷) − (𝐵 + 𝐶))) | ||
Theorem | sub4d 11619 | Rearrangement of 4 terms in a subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) − (𝐶 − 𝐷)) = ((𝐴 − 𝐶) − (𝐵 − 𝐷))) | ||
Theorem | 2addsubd 11620 | Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → (((𝐴 + 𝐵) + 𝐶) − 𝐷) = (((𝐴 + 𝐶) − 𝐷) + 𝐵)) | ||
Theorem | addsubeq4d 11621 | Relation between sums and differences. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐶 − 𝐴) = (𝐵 − 𝐷))) | ||
Theorem | subeqxfrd 11622 | Transfer two terms of a subtraction in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐵 − 𝐷)) | ||
Theorem | mvlraddd 11623 | Move the right term in a sum on the LHS to the RHS, deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = (𝐶 − 𝐵)) | ||
Theorem | mvlladdd 11624 | Move the left term in a sum on the LHS to the RHS, deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶) ⇒ ⊢ (𝜑 → 𝐵 = (𝐶 − 𝐴)) | ||
Theorem | mvrraddd 11625 | Move the right term in a sum on the RHS to the LHS, deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) = 𝐵) | ||
Theorem | mvrladdd 11626 | Move the left term in a sum on the RHS to the LHS, deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶) | ||
Theorem | assraddsubd 11627 | Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 15-Oct-2018.) |
⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = ((𝐵 + 𝐶) − 𝐷)) ⇒ ⊢ (𝜑 → 𝐴 = (𝐵 + (𝐶 − 𝐷))) | ||
Theorem | subaddeqd 11628 | Transfer two terms of a subtraction to an addition in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐷) = (𝐶 − 𝐵)) | ||
Theorem | addlsub 11629 | Left-subtraction: Subtraction of the left summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) = 𝐶 ↔ 𝐴 = (𝐶 − 𝐵))) | ||
Theorem | addrsub 11630 | Right-subtraction: Subtraction of the right summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) = 𝐶 ↔ 𝐵 = (𝐶 − 𝐴))) | ||
Theorem | subexsub 11631 | A subtraction law: Exchanging the subtrahend and the result of the subtraction. (Contributed by BJ, 6-Jun-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 = (𝐶 − 𝐵) ↔ 𝐵 = (𝐶 − 𝐴))) | ||
Theorem | addid0 11632 | If adding a number to a another number yields the other number, the added number must be 0. This shows that 0 is the unique (right) identity of the complex numbers. (Contributed by AV, 17-Jan-2021.) |
⊢ ((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ) → ((𝑋 + 𝑌) = 𝑋 ↔ 𝑌 = 0)) | ||
Theorem | addn0nid 11633 | Adding a nonzero number to a complex number does not yield the complex number. (Contributed by AV, 17-Jan-2021.) |
⊢ ((𝑋 ∈ ℂ ∧ 𝑌 ∈ ℂ ∧ 𝑌 ≠ 0) → (𝑋 + 𝑌) ≠ 𝑋) | ||
Theorem | pnpncand 11634 | Addition/subtraction cancellation law. (Contributed by Scott Fenton, 14-Dec-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + (𝐵 − 𝐶)) + (𝐶 − 𝐵)) = 𝐴) | ||
Theorem | subeqrev 11635 | Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 − 𝐵) = (𝐶 − 𝐷) ↔ (𝐵 − 𝐴) = (𝐷 − 𝐶))) | ||
Theorem | addeq0 11636 | Two complex numbers add up to zero iff they are each other's opposites. (Contributed by Thierry Arnoux, 2-May-2017.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) = 0 ↔ 𝐴 = -𝐵)) | ||
Theorem | pncan1 11637 | Cancellation law for addition and subtraction with 1. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
⊢ (𝐴 ∈ ℂ → ((𝐴 + 1) − 1) = 𝐴) | ||
Theorem | npcan1 11638 | Cancellation law for subtraction and addition with 1. (Contributed by Alexander van der Vekens, 5-Oct-2018.) |
⊢ (𝐴 ∈ ℂ → ((𝐴 − 1) + 1) = 𝐴) | ||
Theorem | subeq0bd 11639 | If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 11580. Converse of subeq0d 11578. Contrapositive of subne0ad 11581. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) = 0) | ||
Theorem | renegcld 11640 | Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → -𝐴 ∈ ℝ) | ||
Theorem | resubcld 11641 | Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ) | ||
Theorem | negn0 11642* | The image under negation of a nonempty set of reals is nonempty. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} ≠ ∅) | ||
Theorem | negf1o 11643* | Negation is an isomorphism of a subset of the real numbers to the negated elements of the subset. (Contributed by AV, 9-Aug-2020.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ -𝑥) ⇒ ⊢ (𝐴 ⊆ ℝ → 𝐹:𝐴–1-1-onto→{𝑛 ∈ ℝ ∣ -𝑛 ∈ 𝐴}) | ||
Theorem | kcnktkm1cn 11644 | k times k minus 1 is a complex number if k is a complex number. (Contributed by Alexander van der Vekens, 11-Mar-2018.) |
⊢ (𝐾 ∈ ℂ → (𝐾 · (𝐾 − 1)) ∈ ℂ) | ||
Theorem | muladd 11645 | Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) + ((𝐴 · 𝐷) + (𝐶 · 𝐵)))) | ||
Theorem | subdi 11646 | Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶))) | ||
Theorem | subdir 11647 | Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶))) | ||
Theorem | ine0 11648 | The imaginary unit i is not zero. (Contributed by NM, 6-May-1999.) |
⊢ i ≠ 0 | ||
Theorem | mulneg1 11649 | Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = -(𝐴 · 𝐵)) | ||
Theorem | mulneg2 11650 | The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) | ||
Theorem | mulneg12 11651 | Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = (𝐴 · -𝐵)) | ||
Theorem | mul2neg 11652 | Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 30-Jul-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) | ||
Theorem | submul2 11653 | Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 · 𝐶)) = (𝐴 + (𝐵 · -𝐶))) | ||
Theorem | mulm1 11654 | Product with minus one is negative. (Contributed by NM, 16-Nov-1999.) |
⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) | ||
Theorem | addneg1mul 11655 | Addition with product with minus one is a subtraction. (Contributed by AV, 18-Oct-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (-1 · 𝐵)) = (𝐴 − 𝐵)) | ||
Theorem | mulsub 11656 | Product of two differences. (Contributed by NM, 14-Jan-2006.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 − 𝐵) · (𝐶 − 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) − ((𝐴 · 𝐷) + (𝐶 · 𝐵)))) | ||
Theorem | mulsub2 11657 | Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 − 𝐵) · (𝐶 − 𝐷)) = ((𝐵 − 𝐴) · (𝐷 − 𝐶))) | ||
Theorem | mulm1i 11658 | Product with minus one is negative. (Contributed by NM, 31-Jul-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (-1 · 𝐴) = -𝐴 | ||
Theorem | mulneg1i 11659 | Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (-𝐴 · 𝐵) = -(𝐴 · 𝐵) | ||
Theorem | mulneg2i 11660 | Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 · -𝐵) = -(𝐴 · 𝐵) | ||
Theorem | mul2negi 11661 | Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (-𝐴 · -𝐵) = (𝐴 · 𝐵) | ||
Theorem | subdii 11662 | Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶)) | ||
Theorem | subdiri 11663 | Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶)) | ||
Theorem | muladdi 11664 | Product of two sums. (Contributed by NM, 17-May-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) + ((𝐴 · 𝐷) + (𝐶 · 𝐵))) | ||
Theorem | mulm1d 11665 | Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (-1 · 𝐴) = -𝐴) | ||
Theorem | mulneg1d 11666 | Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (-𝐴 · 𝐵) = -(𝐴 · 𝐵)) | ||
Theorem | mulneg2d 11667 | Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) | ||
Theorem | mul2negd 11668 | Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (-𝐴 · -𝐵) = (𝐴 · 𝐵)) | ||
Theorem | subdid 11669 | Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · (𝐵 − 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶))) | ||
Theorem | subdird 11670 | Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶))) | ||
Theorem | muladdd 11671 | Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) + ((𝐴 · 𝐷) + (𝐶 · 𝐵)))) | ||
Theorem | mulsubd 11672 | Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) · (𝐶 − 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) − ((𝐴 · 𝐷) + (𝐶 · 𝐵)))) | ||
Theorem | muls1d 11673 | Multiplication by one minus a number. (Contributed by Scott Fenton, 23-Dec-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · (𝐵 − 1)) = ((𝐴 · 𝐵) − 𝐴)) | ||
Theorem | mulsubfacd 11674 | Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) − 𝐵) = ((𝐴 − 1) · 𝐵)) | ||
Theorem | addmulsub 11675 | The product of a sum and a difference. (Contributed by AV, 5-Mar-2023.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) · (𝐶 − 𝐷)) = (((𝐴 · 𝐶) + (𝐵 · 𝐶)) − ((𝐴 · 𝐷) + (𝐵 · 𝐷)))) | ||
Theorem | subaddmulsub 11676 | The difference with a product of a sum and a difference. (Contributed by AV, 5-Mar-2023.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ 𝐸 ∈ ℂ) → (𝐸 − ((𝐴 + 𝐵) · (𝐶 − 𝐷))) = (((𝐸 − (𝐴 · 𝐶)) − (𝐵 · 𝐶)) + ((𝐴 · 𝐷) + (𝐵 · 𝐷)))) | ||
Theorem | mulsubaddmulsub 11677 | A special difference of a product with a product of a sum and a difference. (Contributed by AV, 5-Mar-2023.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐵 · 𝐶) − ((𝐴 + 𝐵) · (𝐶 − 𝐷))) = (((𝐴 · 𝐷) + (𝐵 · 𝐷)) − (𝐴 · 𝐶))) | ||
Theorem | gt0ne0 11678 | Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) | ||
Theorem | lt0ne0 11679 | A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ≠ 0) | ||
Theorem | ltadd1 11680 | Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶))) | ||
Theorem | leadd1 11681 | Addition to both sides of 'less than or equal to'. (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶))) | ||
Theorem | leadd2 11682 | Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))) | ||
Theorem | ltsubadd 11683 | 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 + 𝐵))) | ||
Theorem | ltsubadd2 11684 | 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐵 + 𝐶))) | ||
Theorem | lesubadd 11685 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵))) | ||
Theorem | lesubadd2 11686 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐵 + 𝐶))) | ||
Theorem | ltaddsub 11687 | 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 − 𝐵))) | ||
Theorem | ltaddsub2 11688 | 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐵 < (𝐶 − 𝐴))) | ||
Theorem | leaddsub 11689 | 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 − 𝐵))) | ||
Theorem | leaddsub2 11690 | 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶 ↔ 𝐵 ≤ (𝐶 − 𝐴))) | ||
Theorem | suble 11691 | Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 𝐶 ↔ (𝐴 − 𝐶) ≤ 𝐵)) | ||
Theorem | lesub 11692 | Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ (𝐵 − 𝐶) ↔ 𝐶 ≤ (𝐵 − 𝐴))) | ||
Theorem | ltsub23 11693 | 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) < 𝐶 ↔ (𝐴 − 𝐶) < 𝐵)) | ||
Theorem | ltsub13 11694 | 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < (𝐵 − 𝐶) ↔ 𝐶 < (𝐵 − 𝐴))) | ||
Theorem | le2add 11695 | Adding both sides of two 'less than or equal to' relations. (Contributed by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))) | ||
Theorem | ltleadd 11696 | Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 < 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷))) | ||
Theorem | leltadd 11697 | Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷))) | ||
Theorem | lt2add 11698 | Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 < 𝐶 ∧ 𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷))) | ||
Theorem | addgt0 11699 | The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 < 𝐵)) → 0 < (𝐴 + 𝐵)) | ||
Theorem | addgegt0 11700 | The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 < 𝐵)) → 0 < (𝐴 + 𝐵)) |
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