Theorem List for Intuitionistic Logic Explorer - 8201-8300 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | subid 8201 |
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 8202 |
Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised
by Mario Carneiro, 27-May-2016.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴) |
|
Theorem | npncan 8203 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐵 − 𝐶)) = (𝐴 − 𝐶)) |
|
Theorem | nppcan 8204 |
Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 − 𝐵) + 𝐶) + 𝐵) = (𝐴 + 𝐶)) |
|
Theorem | nnpcan 8205 |
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 8206 |
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 8207 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵)) |
|
Theorem | subeq0 8208 |
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 8209 |
Cancellation law for subtraction. (Contributed by Scott Fenton,
21-Jun-2013.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐵 − 𝐴)) = 0) |
|
Theorem | subsub2 8210 |
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 8211 |
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 8212 |
Law for double subtraction. (Contributed by NM, 13-May-2004.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 − 𝐵) + 𝐶)) |
|
Theorem | nppcan2 8213 |
Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 + 𝐶)) + 𝐶) = (𝐴 − 𝐵)) |
|
Theorem | subsub3 8214 |
Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 + 𝐶) − 𝐵)) |
|
Theorem | subsub4 8215 |
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 8216 |
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 8217 |
Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 − 𝐶)) − 𝐶) = (𝐴 − 𝐵)) |
|
Theorem | nnncan1 8218 |
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 8219 |
Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶) − (𝐵 − 𝐶)) = (𝐴 − 𝐵)) |
|
Theorem | npncan3 8220 |
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 8221 |
Cancellation law for mixed addition and subtraction. (Contributed by NM,
4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵 − 𝐶)) |
|
Theorem | pnpcan2 8222 |
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 8223 |
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 8224 |
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 8225 |
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 8226 |
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 8227 |
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 8228 |
Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
|
⊢ -0 = 0 |
|
Theorem | negid 8229 |
Addition of a number and its negative. (Contributed by NM,
14-Mar-2005.)
|
⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) |
|
Theorem | negsub 8230 |
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 8231 |
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 8232 |
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 8233 |
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 8234 |
Negative contraposition law. (Contributed by NM, 9-May-2004.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)) |
|
Theorem | negcon2 8235 |
Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = -𝐵 ↔ 𝐵 = -𝐴)) |
|
Theorem | negeq0 8236 |
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 8237 |
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 8238 |
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 8239 |
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 8240 |
Distribution of negative over addition. (Contributed by NM,
1-Jan-2006.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 − 𝐵)) |
|
Theorem | negsubdi2 8241 |
Distribution of negative over subtraction. (Contributed by NM,
4-Oct-1999.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) |
|
Theorem | neg2sub 8242 |
Relationship between subtraction and negative. (Contributed by Paul
Chapman, 8-Oct-2007.)
|
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 − -𝐵) = (𝐵 − 𝐴)) |
|
Theorem | renegcl 8243 |
Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
|
⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) |
|
Theorem | renegcli 8244 |
Closure law for negative of reals. (Note: this inference proof style
and the deduction theorem usage in renegcl 8243 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 8245 |
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 | resubcl 8246 |
Closure law for subtraction of reals. (Contributed by NM,
20-Jan-1997.)
|
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) |
|
Theorem | negreb 8247 |
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 8248 |
"Reverse" second Peano postulate analog 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 8249 |
"Reverse" second Peano postulate analog for reals. (Contributed by
NM,
6-Feb-2007.)
|
⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈
ℝ) |
|
Theorem | negcli 8250 |
Closure law for negative. (Contributed by NM, 26-Nov-1994.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ -𝐴 ∈ ℂ |
|
Theorem | negidi 8251 |
Addition of a number and its negative. (Contributed by NM,
26-Nov-1994.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 + -𝐴) = 0 |
|
Theorem | negnegi 8252 |
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 8253 |
Subtraction of a number from itself. (Contributed by NM,
26-Nov-1994.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 − 𝐴) = 0 |
|
Theorem | subid1i 8254 |
Identity law for subtraction. (Contributed by NM, 29-May-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 − 0) = 𝐴 |
|
Theorem | negne0bi 8255 |
A number is nonzero iff its negative is nonzero. (Contributed by NM,
10-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (𝐴 ≠ 0 ↔ -𝐴 ≠ 0) |
|
Theorem | negrebi 8256 |
The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ
⇒ ⊢ (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ) |
|
Theorem | negne0i 8257 |
The negative of a nonzero number is nonzero. (Contributed by NM,
30-Jul-2004.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠
0 ⇒ ⊢ -𝐴 ≠ 0 |
|
Theorem | subcli 8258 |
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 8259 |
Subtraction and addition of equals. (Contributed by NM,
26-Nov-1994.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐴 + (𝐵 − 𝐴)) = 𝐵 |
|
Theorem | negsubi 8260 |
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 8261 |
Relationship between subtraction and negative. (Contributed by NM,
1-Dec-2005.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐴 − -𝐵) = (𝐴 + 𝐵) |
|
Theorem | subeq0i 8262 |
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 8263 |
Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵) |
|
Theorem | negcon1i 8264 |
Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴) |
|
Theorem | negcon2i 8265 |
Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ (𝐴 = -𝐵 ↔ 𝐵 = -𝐴) |
|
Theorem | negdii 8266 |
Distribution of negative over addition. (Contributed by NM,
28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ -(𝐴 + 𝐵) = (-𝐴 + -𝐵) |
|
Theorem | negsubdii 8267 |
Distribution of negative over subtraction. (Contributed by NM,
6-Aug-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ -(𝐴 − 𝐵) = (-𝐴 + 𝐵) |
|
Theorem | negsubdi2i 8268 |
Distribution of negative over subtraction. (Contributed by NM,
1-Oct-1999.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈
ℂ ⇒ ⊢ -(𝐴 − 𝐵) = (𝐵 − 𝐴) |
|
Theorem | subaddi 8269 |
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 8270 |
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 8271 |
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 8272 |
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 8273 |
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 8274 |
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 8275 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 − 𝐵) = (𝐴 − 𝐶) ↔ 𝐵 = 𝐶) |
|
Theorem | subcan2i 8276 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈
ℂ ⇒ ⊢ ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵) |
|
Theorem | pnncani 8277 |
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 8278 |
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 8279 |
Alternate proof of 0re 7982. (Contributed by NM, 19-Feb-2005.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ 0 ∈ ℝ |
|
Theorem | negcld 8280 |
Closure law for negative. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → -𝐴 ∈ ℂ) |
|
Theorem | subidd 8281 |
Subtraction of a number from itself. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
|
Theorem | subid1d 8282 |
Identity law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 − 0) = 𝐴) |
|
Theorem | negidd 8283 |
Addition of a number and its negative. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 + -𝐴) = 0) |
|
Theorem | negnegd 8284 |
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 8285 |
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 8286 |
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 8287 |
Contraposition law for unary minus. Deduction form of negcon1 8234.
(Contributed by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)) |
|
Theorem | negcon1ad 8288 |
Contraposition law for unary minus. One-way deduction form of
negcon1 8234. (Contributed by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → -𝐴 = 𝐵) ⇒ ⊢ (𝜑 → -𝐵 = 𝐴) |
|
Theorem | neg11ad 8289 |
The negatives of two complex numbers are equal iff they are equal.
Deduction form of neg11 8233. Generalization of neg11d 8305.
(Contributed by David Moews, 28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵)) |
|
Theorem | negned 8290 |
If two complex numbers are unequal, so are their negatives.
Contrapositive of neg11d 8305. (Contributed by David Moews,
28-Feb-2017.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → -𝐴 ≠ -𝐵) |
|
Theorem | negne0d 8291 |
The negative of a nonzero number is nonzero. See also negap0d 8613 which
is similar but for apart from zero rather than not equal to zero.
(Contributed by Mario Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → -𝐴 ≠ 0) |
|
Theorem | negrebd 8292 |
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 8293 |
Closure law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
|
Theorem | pncand 8294 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
|
Theorem | pncan2d 8295 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐴) = 𝐵) |
|
Theorem | pncan3d 8296 |
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 8297 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
|
Theorem | nncand 8298 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) = 𝐵) |
|
Theorem | negsubd 8299 |
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 8300 |
Relationship between subtraction and negative. (Contributed by Mario
Carneiro, 27-May-2016.)
|
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → (𝐴 − -𝐵) = (𝐴 + 𝐵)) |