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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | addsub12 10901 | Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 − 𝐶)) = (𝐵 + (𝐴 − 𝐶))) | ||
Theorem | 2addsub 10902 | Law for subtraction and addition. (Contributed by NM, 20-Nov-2005.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (((𝐴 + 𝐵) + 𝐶) − 𝐷) = (((𝐴 + 𝐶) − 𝐷) + 𝐵)) | ||
Theorem | addsubeq4 10903 | Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐶 − 𝐴) = (𝐵 − 𝐷))) | ||
Theorem | pncan3oi 10904 | Subtraction and addition of equals. Almost but not exactly the same as pncan3i 10965 and pncan 10894, this order happens often when applying "operations to both sides" so create a theorem specifically for it. A deduction version of this is available as pncand 11000. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) − 𝐵) = 𝐴 | ||
Theorem | mvrraddi 10905 | Move RHS right addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐴 = (𝐵 + 𝐶) ⇒ ⊢ (𝐴 − 𝐶) = 𝐵 | ||
Theorem | mvlladdi 10906 | Move LHS left addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ (𝐴 + 𝐵) = 𝐶 ⇒ ⊢ 𝐵 = (𝐶 − 𝐴) | ||
Theorem | subid 10907 | Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (𝐴 ∈ ℂ → (𝐴 − 𝐴) = 0) | ||
Theorem | subid1 10908 | Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴) | ||
Theorem | npncan 10909 | Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐵 − 𝐶)) = (𝐴 − 𝐶)) | ||
Theorem | nppcan 10910 | Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 − 𝐵) + 𝐶) + 𝐵) = (𝐴 + 𝐶)) | ||
Theorem | nnpcan 10911 | Cancellation law for subtraction: ((a-b)-c)+b = a-c holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 24-Mar-2018.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 − 𝐵) − 𝐶) + 𝐵) = (𝐴 − 𝐶)) | ||
Theorem | nppcan3 10912 | Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐶 + 𝐵)) = (𝐴 + 𝐶)) | ||
Theorem | subcan2 10913 | Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | subeq0 10914 | If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) | ||
Theorem | npncan2 10915 | Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐵 − 𝐴)) = 0) | ||
Theorem | subsub2 10916 | Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = (𝐴 + (𝐶 − 𝐵))) | ||
Theorem | nncan 10917 | Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (𝐴 − 𝐵)) = 𝐵) | ||
Theorem | subsub 10918 | Law for double subtraction. (Contributed by NM, 13-May-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 − 𝐵) + 𝐶)) | ||
Theorem | nppcan2 10919 | Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 + 𝐶)) + 𝐶) = (𝐴 − 𝐵)) | ||
Theorem | subsub3 10920 | Law for double subtraction. (Contributed by NM, 27-Jul-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 + 𝐶) − 𝐵)) | ||
Theorem | subsub4 10921 | Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) − 𝐶) = (𝐴 − (𝐵 + 𝐶))) | ||
Theorem | sub32 10922 | Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) − 𝐶) = ((𝐴 − 𝐶) − 𝐵)) | ||
Theorem | nnncan 10923 | Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 − 𝐶)) − 𝐶) = (𝐴 − 𝐵)) | ||
Theorem | nnncan1 10924 | Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) − (𝐴 − 𝐶)) = (𝐶 − 𝐵)) | ||
Theorem | nnncan2 10925 | Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶) − (𝐵 − 𝐶)) = (𝐴 − 𝐵)) | ||
Theorem | npncan3 10926 | Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐶 − 𝐴)) = (𝐶 − 𝐵)) | ||
Theorem | pnpcan 10927 | Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by SN, 13-Nov-2023.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵 − 𝐶)) | ||
Theorem | pnpcan2 10928 | Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐶) − (𝐵 + 𝐶)) = (𝐴 − 𝐵)) | ||
Theorem | pnncan 10929 | Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − (𝐴 − 𝐶)) = (𝐵 + 𝐶)) | ||
Theorem | ppncan 10930 | Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + (𝐶 − 𝐵)) = (𝐴 + 𝐶)) | ||
Theorem | addsub4 10931 | Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) − (𝐶 + 𝐷)) = ((𝐴 − 𝐶) + (𝐵 − 𝐷))) | ||
Theorem | subadd4 10932 | Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 − 𝐵) − (𝐶 − 𝐷)) = ((𝐴 + 𝐷) − (𝐵 + 𝐶))) | ||
Theorem | sub4 10933 | Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 − 𝐵) − (𝐶 − 𝐷)) = ((𝐴 − 𝐶) − (𝐵 − 𝐷))) | ||
Theorem | neg0 10934 | Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.) |
⊢ -0 = 0 | ||
Theorem | negid 10935 | Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.) |
⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) | ||
Theorem | negsub 10936 | Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | ||
Theorem | subneg 10937 | Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − -𝐵) = (𝐴 + 𝐵)) | ||
Theorem | negneg 10938 | A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 12-Jan-2002.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) | ||
Theorem | neg11 10939 | Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | negcon1 10940 | Negative contraposition law. (Contributed by NM, 9-May-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)) | ||
Theorem | negcon2 10941 | Negative contraposition law. (Contributed by NM, 14-Nov-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = -𝐵 ↔ 𝐵 = -𝐴)) | ||
Theorem | negeq0 10942 | A number is zero iff its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (𝐴 ∈ ℂ → (𝐴 = 0 ↔ -𝐴 = 0)) | ||
Theorem | subcan 10943 | Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) = (𝐴 − 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | negsubdi 10944 | Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (-𝐴 + 𝐵)) | ||
Theorem | negdi 10945 | Distribution of negative over addition. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵)) | ||
Theorem | negdi2 10946 | Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 − 𝐵)) | ||
Theorem | negsubdi2 10947 | Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) | ||
Theorem | neg2sub 10948 | Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 − -𝐵) = (𝐵 − 𝐴)) | ||
Theorem | renegcli 10949 | Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 10951 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ -𝐴 ∈ ℝ | ||
Theorem | resubcli 10950 | Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 − 𝐵) ∈ ℝ | ||
Theorem | renegcl 10951 | Closure law for negative of reals. The weak deduction theorem dedth 4525 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 10949, to an antecedent. (Contributed by NM, 20-Jan-1997.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | ||
Theorem | resubcl 10952 | Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) | ||
Theorem | negreb 10953 | The negative of a real is real. (Contributed by NM, 11-Aug-1999.) (Revised by Mario Carneiro, 14-Jul-2014.) |
⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)) | ||
Theorem | peano2cnm 10954 | "Reverse" second Peano postulate analogue for complex numbers: A complex number minus 1 is a complex number. (Contributed by Alexander van der Vekens, 18-Mar-2018.) |
⊢ (𝑁 ∈ ℂ → (𝑁 − 1) ∈ ℂ) | ||
Theorem | peano2rem 10955 | "Reverse" second Peano postulate analogue for reals. (Contributed by NM, 6-Feb-2007.) |
⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ) | ||
Theorem | negcli 10956 | Closure law for negative. (Contributed by NM, 26-Nov-1994.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ -𝐴 ∈ ℂ | ||
Theorem | negidi 10957 | Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 + -𝐴) = 0 | ||
Theorem | negnegi 10958 | A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 8-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ --𝐴 = 𝐴 | ||
Theorem | subidi 10959 | Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 − 𝐴) = 0 | ||
Theorem | subid1i 10960 | Identity law for subtraction. (Contributed by NM, 29-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 − 0) = 𝐴 | ||
Theorem | negne0bi 10961 | A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 ≠ 0 ↔ -𝐴 ≠ 0) | ||
Theorem | negrebi 10962 | The negative of a real is real. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ) | ||
Theorem | negne0i 10963 | The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠ 0 ⇒ ⊢ -𝐴 ≠ 0 | ||
Theorem | subcli 10964 | Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 − 𝐵) ∈ ℂ | ||
Theorem | pncan3i 10965 | Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 + (𝐵 − 𝐴)) = 𝐵 | ||
Theorem | negsubi 10966 | Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 + -𝐵) = (𝐴 − 𝐵) | ||
Theorem | subnegi 10967 | Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 − -𝐵) = (𝐴 + 𝐵) | ||
Theorem | subeq0i 10968 | If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵) | ||
Theorem | neg11i 10969 | Negative is one-to-one. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵) | ||
Theorem | negcon1i 10970 | Negative contraposition law. (Contributed by NM, 25-Aug-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴) | ||
Theorem | negcon2i 10971 | Negative contraposition law. (Contributed by NM, 25-Aug-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 = -𝐵 ↔ 𝐵 = -𝐴) | ||
Theorem | negdii 10972 | Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by OpenAI, 25-Mar-2011.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ -(𝐴 + 𝐵) = (-𝐴 + -𝐵) | ||
Theorem | negsubdii 10973 | Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ -(𝐴 − 𝐵) = (-𝐴 + 𝐵) | ||
Theorem | negsubdi2i 10974 | Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ -(𝐴 − 𝐵) = (𝐵 − 𝐴) | ||
Theorem | subaddi 10975 | Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 − 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴) | ||
Theorem | subadd2i 10976 | Relationship between subtraction and addition. (Contributed by NM, 15-Dec-2006.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 − 𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴) | ||
Theorem | subaddrii 10977 | Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ (𝐵 + 𝐶) = 𝐴 ⇒ ⊢ (𝐴 − 𝐵) = 𝐶 | ||
Theorem | subsub23i 10978 | Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 − 𝐵) = 𝐶 ↔ (𝐴 − 𝐶) = 𝐵) | ||
Theorem | addsubassi 10979 | Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵 − 𝐶)) | ||
Theorem | addsubi 10980 | Law for subtraction and addition. (Contributed by NM, 6-Aug-2003.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵) | ||
Theorem | subcani 10981 | Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 − 𝐵) = (𝐴 − 𝐶) ↔ 𝐵 = 𝐶) | ||
Theorem | subcan2i 10982 | Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵) | ||
Theorem | pnncani 10983 | Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) − (𝐴 − 𝐶)) = (𝐵 + 𝐶) | ||
Theorem | addsub4i 10984 | Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) − (𝐶 + 𝐷)) = ((𝐴 − 𝐶) + (𝐵 − 𝐷)) | ||
Theorem | 0reALT 10985 | Alternate proof of 0re 10645. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 0 ∈ ℝ | ||
Theorem | negcld 10986 | Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → -𝐴 ∈ ℂ) | ||
Theorem | subidd 10987 | Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 − 𝐴) = 0) | ||
Theorem | subid1d 10988 | Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 − 0) = 𝐴) | ||
Theorem | negidd 10989 | Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 + -𝐴) = 0) | ||
Theorem | negnegd 10990 | A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → --𝐴 = 𝐴) | ||
Theorem | negeq0d 10991 | A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 = 0 ↔ -𝐴 = 0)) | ||
Theorem | negne0bd 10992 | A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 ≠ 0 ↔ -𝐴 ≠ 0)) | ||
Theorem | negcon1d 10993 | Contraposition law for unary minus. Deduction form of negcon1 10940. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)) | ||
Theorem | negcon1ad 10994 | Contraposition law for unary minus. One-way deduction form of negcon1 10940. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → -𝐴 = 𝐵) ⇒ ⊢ (𝜑 → -𝐵 = 𝐴) | ||
Theorem | neg11ad 10995 | The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 10939. Generalization of neg11d 11011. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | negned 10996 | If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 11011. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → -𝐴 ≠ -𝐵) | ||
Theorem | negne0d 10997 | The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → -𝐴 ≠ 0) | ||
Theorem | negrebd 10998 | The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → -𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | subcld 10999 | Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) | ||
Theorem | pncand 11000 | Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
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