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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | npncan 10901 | Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐵 − 𝐶)) = (𝐴 − 𝐶)) | ||
Theorem | nppcan 10902 | Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 − 𝐵) + 𝐶) + 𝐵) = (𝐴 + 𝐶)) | ||
Theorem | nnpcan 10903 | 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 10904 | 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 10905 | Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | subeq0 10906 | 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 10907 | Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐵 − 𝐴)) = 0) | ||
Theorem | subsub2 10908 | 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 10909 | 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 10910 | Law for double subtraction. (Contributed by NM, 13-May-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 − 𝐵) + 𝐶)) | ||
Theorem | nppcan2 10911 | Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 + 𝐶)) + 𝐶) = (𝐴 − 𝐵)) | ||
Theorem | subsub3 10912 | Law for double subtraction. (Contributed by NM, 27-Jul-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 + 𝐶) − 𝐵)) | ||
Theorem | subsub4 10913 | 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 10914 | 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 10915 | Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 − 𝐶)) − 𝐶) = (𝐴 − 𝐵)) | ||
Theorem | nnncan1 10916 | 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 10917 | Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶) − (𝐵 − 𝐶)) = (𝐴 − 𝐵)) | ||
Theorem | npncan3 10918 | 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 10919 | 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 10920 | 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 10921 | 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 10922 | 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 10923 | 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 10924 | 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 10925 | 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 10926 | Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.) |
⊢ -0 = 0 | ||
Theorem | negid 10927 | Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.) |
⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) | ||
Theorem | negsub 10928 | 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 10929 | 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 10930 | 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 10931 | 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 10932 | Negative contraposition law. (Contributed by NM, 9-May-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)) | ||
Theorem | negcon2 10933 | Negative contraposition law. (Contributed by NM, 14-Nov-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = -𝐵 ↔ 𝐵 = -𝐴)) | ||
Theorem | negeq0 10934 | 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 10935 | 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 10936 | 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 10937 | 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 10938 | Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 − 𝐵)) | ||
Theorem | negsubdi2 10939 | Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) | ||
Theorem | neg2sub 10940 | Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 − -𝐵) = (𝐵 − 𝐴)) | ||
Theorem | renegcli 10941 | Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 10943 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 10942 | 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 10943 | Closure law for negative of reals. The weak deduction theorem dedth 4523 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 10941, to an antecedent. (Contributed by NM, 20-Jan-1997.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | ||
Theorem | resubcl 10944 | Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) | ||
Theorem | negreb 10945 | 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 10946 | "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 10947 | "Reverse" second Peano postulate analogue for reals. (Contributed by NM, 6-Feb-2007.) |
⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ) | ||
Theorem | negcli 10948 | Closure law for negative. (Contributed by NM, 26-Nov-1994.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ -𝐴 ∈ ℂ | ||
Theorem | negidi 10949 | Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 + -𝐴) = 0 | ||
Theorem | negnegi 10950 | 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 10951 | Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 − 𝐴) = 0 | ||
Theorem | subid1i 10952 | Identity law for subtraction. (Contributed by NM, 29-May-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 − 0) = 𝐴 | ||
Theorem | negne0bi 10953 | A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 ≠ 0 ↔ -𝐴 ≠ 0) | ||
Theorem | negrebi 10954 | The negative of a real is real. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ) | ||
Theorem | negne0i 10955 | The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠ 0 ⇒ ⊢ -𝐴 ≠ 0 | ||
Theorem | subcli 10956 | 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 10957 | Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 + (𝐵 − 𝐴)) = 𝐵 | ||
Theorem | negsubi 10958 | 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 10959 | Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 − -𝐵) = (𝐴 + 𝐵) | ||
Theorem | subeq0i 10960 | 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 10961 | Negative is one-to-one. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵) | ||
Theorem | negcon1i 10962 | Negative contraposition law. (Contributed by NM, 25-Aug-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴) | ||
Theorem | negcon2i 10963 | Negative contraposition law. (Contributed by NM, 25-Aug-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 = -𝐵 ↔ 𝐵 = -𝐴) | ||
Theorem | negdii 10964 | 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 10965 | Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ -(𝐴 − 𝐵) = (-𝐴 + 𝐵) | ||
Theorem | negsubdi2i 10966 | Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ -(𝐴 − 𝐵) = (𝐵 − 𝐴) | ||
Theorem | subaddi 10967 | 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 10968 | 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 10969 | 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 10970 | 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 10971 | 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 10972 | 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 10973 | Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 − 𝐵) = (𝐴 − 𝐶) ↔ 𝐵 = 𝐶) | ||
Theorem | subcan2i 10974 | Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵) | ||
Theorem | pnncani 10975 | 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 10976 | 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 10977 | Alternate proof of 0re 10637. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 0 ∈ ℝ | ||
Theorem | negcld 10978 | Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → -𝐴 ∈ ℂ) | ||
Theorem | subidd 10979 | Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 − 𝐴) = 0) | ||
Theorem | subid1d 10980 | Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 − 0) = 𝐴) | ||
Theorem | negidd 10981 | Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 + -𝐴) = 0) | ||
Theorem | negnegd 10982 | 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 10983 | 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 10984 | 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 10985 | Contraposition law for unary minus. Deduction form of negcon1 10932. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)) | ||
Theorem | negcon1ad 10986 | Contraposition law for unary minus. One-way deduction form of negcon1 10932. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → -𝐴 = 𝐵) ⇒ ⊢ (𝜑 → -𝐵 = 𝐴) | ||
Theorem | neg11ad 10987 | The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 10931. Generalization of neg11d 11003. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | negned 10988 | If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 11003. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → -𝐴 ≠ -𝐵) | ||
Theorem | negne0d 10989 | 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 10990 | 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 10991 | Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) | ||
Theorem | pncand 10992 | Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴) | ||
Theorem | pncan2d 10993 | Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐴) = 𝐵) | ||
Theorem | pncan3d 10994 | Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵) | ||
Theorem | npcand 10995 | Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) = 𝐴) | ||
Theorem | nncand 10996 | Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) = 𝐵) | ||
Theorem | negsubd 10997 | Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | ||
Theorem | subnegd 10998 | Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 − -𝐵) = (𝐴 + 𝐵)) | ||
Theorem | subeq0d 10999 | If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 − 𝐵) = 0) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | subne0d 11000 | Two unequal numbers have nonzero difference. (Contributed by Mario Carneiro, 1-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0) |
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