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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | negsubdid 11001 | Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → -(𝐴 − 𝐵) = (-𝐴 + 𝐵)) | ||
Theorem | negsubdi2d 11002 | Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) | ||
Theorem | neg2subd 11003 | Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (-𝐴 − -𝐵) = (𝐵 − 𝐴)) | ||
Theorem | subaddd 11004 | Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴)) | ||
Theorem | subadd2d 11005 | Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴)) | ||
Theorem | addsubassd 11006 | Associative-type law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵 − 𝐶))) | ||
Theorem | addsubd 11007 | Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵)) | ||
Theorem | subadd23d 11008 | Commutative/associative law for addition and 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 | addsub12d 11009 | Commutative/associative law for 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 | npncand 11010 | Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) + (𝐵 − 𝐶)) = (𝐴 − 𝐶)) | ||
Theorem | nppcand 11011 | Cancellation law for 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 | nppcan2d 11012 | 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 | nppcan3d 11013 | Cancellation law for 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 | subsubd 11014 | 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 | subsub2d 11015 | 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 + (𝐶 − 𝐵))) | ||
Theorem | subsub3d 11016 | 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 + 𝐶) − 𝐵)) | ||
Theorem | subsub4d 11017 | 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 11018 | 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 11019 | 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 11020 | 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 11021 | 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 11022 | 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 11023 | 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 11024 | 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 11025 | 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 11026 | 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 11027 | 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 11028 | 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 11029 | Cancellation law for subtraction. Deduction form of subcan 10930. Generalization of subcand 11027. (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 11030 | Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand 11027. (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 11031 | Cancellation law for subtraction. Deduction form of subcan2 10900. Generalization of subcan2d 11028. (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 11032 | Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 11028. (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 11033 | 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 11034 | 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 11035 | 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 11036 | 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 11037 | 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 11038 | 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 11039 | Move LHS right addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = (𝐶 − 𝐵)) | ||
Theorem | mvlladdd 11040 | Move LHS left addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶) ⇒ ⊢ (𝜑 → 𝐵 = (𝐶 − 𝐴)) | ||
Theorem | mvrraddd 11041 | Move RHS right addition to LHS. (Contributed by David A. Wheeler, 15-Oct-2018.) |
⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) = 𝐵) | ||
Theorem | mvrladdd 11042 | Move RHS left addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶) | ||
Theorem | assraddsubd 11043 | 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 11044 | 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 11045 | 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 11046 | 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 11047 | 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 11048 | 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 11049 | 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 11050 | 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 11051 | 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 11052 | 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 11053 | 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 11054 | 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 11055 | If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 10996. Converse of subeq0d 10994. Contrapositive of subne0ad 10997. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) = 0) | ||
Theorem | renegcld 11056 | Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → -𝐴 ∈ ℝ) | ||
Theorem | resubcld 11057 | 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 11058* | 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 11059* | 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 11060 | 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 11061 | 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 11062 | 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 11063 | 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 11064 | The imaginary unit i is not zero. (Contributed by NM, 6-May-1999.) |
⊢ i ≠ 0 | ||
Theorem | mulneg1 11065 | 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 11066 | 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 11067 | Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = (𝐴 · -𝐵)) | ||
Theorem | mul2neg 11068 | 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 11069 | 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 11070 | Product with minus one is negative. (Contributed by NM, 16-Nov-1999.) |
⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) | ||
Theorem | addneg1mul 11071 | 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 11072 | 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 11073 | 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 11074 | Product with minus one is negative. (Contributed by NM, 31-Jul-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (-1 · 𝐴) = -𝐴 | ||
Theorem | mulneg1i 11075 | 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 11076 | 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 11077 | 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 11078 | 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 11079 | 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 11080 | 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 11081 | Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (-1 · 𝐴) = -𝐴) | ||
Theorem | mulneg1d 11082 | 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 11083 | 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 11084 | 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 11085 | 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 11086 | 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 11087 | 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 11088 | 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 11089 | 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 11090 | 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 11091 | 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 11092 | 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 11093 | 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 11094 | 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 11095 | 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 11096 | 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 11097 | 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 11098 | 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 11099 | '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 11100 | '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))) |
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