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Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaddsubass 7201 Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵𝐶)))
 
Theoremaddsub 7202 Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐶) = ((𝐴𝐶) + 𝐵))
 
Theoremsubadd23 7203 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + 𝐶) = (𝐴 + (𝐶𝐵)))
 
Theoremaddsub12 7204 Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵𝐶)) = (𝐵 + (𝐴𝐶)))
 
Theorem2addsub 7205 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))
 
Theoremaddsubeq4 7206 Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐶𝐴) = (𝐵𝐷)))
 
Theorempncan3oi 7207 Subtraction and addition of equals. Almost but not exactly the same as pncan3i 7267 and pncan 7197, 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 7302. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 + 𝐵) − 𝐵) = 𝐴
 
Theoremmvlladdi 7208 Move LHS left addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 + 𝐵) = 𝐶       𝐵 = (𝐶𝐴)
 
Theoremsubid 7209 Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴𝐴) = 0)
 
Theoremsubid1 7210 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)
 
Theoremnpncan 7211 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + (𝐵𝐶)) = (𝐴𝐶))
 
Theoremnppcan 7212 Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴𝐵) + 𝐶) + 𝐵) = (𝐴 + 𝐶))
 
Theoremnnpcan 7213 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) → (((𝐴𝐵) − 𝐶) + 𝐵) = (𝐴𝐶))
 
Theoremnppcan3 7214 Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + (𝐶 + 𝐵)) = (𝐴 + 𝐶))
 
Theoremsubcan2 7215 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐶) = (𝐵𝐶) ↔ 𝐴 = 𝐵))
 
Theoremsubeq0 7216 If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵) = 0 ↔ 𝐴 = 𝐵))
 
Theoremnpncan2 7217 Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵) + (𝐵𝐴)) = 0)
 
Theoremsubsub2 7218 Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵𝐶)) = (𝐴 + (𝐶𝐵)))
 
Theoremnncan 7219 Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (𝐴𝐵)) = 𝐵)
 
Theoremsubsub 7220 Law for double subtraction. (Contributed by NM, 13-May-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵𝐶)) = ((𝐴𝐵) + 𝐶))
 
Theoremnppcan2 7221 Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 + 𝐶)) + 𝐶) = (𝐴𝐵))
 
Theoremsubsub3 7222 Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵𝐶)) = ((𝐴 + 𝐶) − 𝐵))
 
Theoremsubsub4 7223 Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) − 𝐶) = (𝐴 − (𝐵 + 𝐶)))
 
Theoremsub32 7224 Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) − 𝐶) = ((𝐴𝐶) − 𝐵))
 
Theoremnnncan 7225 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵𝐶)) − 𝐶) = (𝐴𝐵))
 
Theoremnnncan1 7226 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) − (𝐴𝐶)) = (𝐶𝐵))
 
Theoremnnncan2 7227 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐶) − (𝐵𝐶)) = (𝐴𝐵))
 
Theoremnpncan3 7228 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) → ((𝐴𝐵) + (𝐶𝐴)) = (𝐶𝐵))
 
Theorempnpcan 7229 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)) = (𝐵𝐶))
 
Theorempnpcan2 7230 Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐶) − (𝐵 + 𝐶)) = (𝐴𝐵))
 
Theorempnncan 7231 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) − (𝐴𝐶)) = (𝐵 + 𝐶))
 
Theoremppncan 7232 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + (𝐶𝐵)) = (𝐴 + 𝐶))
 
Theoremaddsub4 7233 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)) = ((𝐴𝐶) + (𝐵𝐷)))
 
Theoremsubadd4 7234 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴𝐵) − (𝐶𝐷)) = ((𝐴 + 𝐷) − (𝐵 + 𝐶)))
 
Theoremsub4 7235 Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴𝐵) − (𝐶𝐷)) = ((𝐴𝐶) − (𝐵𝐷)))
 
Theoremneg0 7236 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
-0 = 0
 
Theoremnegid 7237 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
(𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0)
 
Theoremnegsub 7238 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) = (𝐴𝐵))
 
Theoremsubneg 7239 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))
 
Theoremnegneg 7240 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)
 
Theoremneg11 7241 Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = -𝐵𝐴 = 𝐵))
 
Theoremnegcon1 7242 Negative contraposition law. (Contributed by NM, 9-May-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
 
Theoremnegcon2 7243 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = -𝐵𝐵 = -𝐴))
 
Theoremnegeq0 7244 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))
 
Theoremsubcan 7245 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = (𝐴𝐶) ↔ 𝐵 = 𝐶))
 
Theoremnegsubdi 7246 Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴𝐵) = (-𝐴 + 𝐵))
 
Theoremnegdi 7247 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))
 
Theoremnegdi2 7248 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴𝐵))
 
Theoremnegsubdi2 7249 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴𝐵) = (𝐵𝐴))
 
Theoremneg2sub 7250 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 − -𝐵) = (𝐵𝐴))
 
Theoremrenegcl 7251 Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
(𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
 
Theoremrenegcli 7252 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 7251 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
 
Theoremresubcli 7253 Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴𝐵) ∈ ℝ
 
Theoremresubcl 7254 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵) ∈ ℝ)
 
Theoremnegreb 7255 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))
 
Theorempeano2cnm 7256 "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)
 
Theorempeano2rem 7257 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)
(𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)
 
Theoremnegcli 7258 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       -𝐴 ∈ ℂ
 
Theoremnegidi 7259 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       (𝐴 + -𝐴) = 0
 
Theoremnegnegi 7260 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
 
Theoremsubidi 7261 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       (𝐴𝐴) = 0
 
Theoremsubid1i 7262 Identity law for subtraction. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 − 0) = 𝐴
 
Theoremnegne0bi 7263 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
𝐴 ∈ ℂ       (𝐴 ≠ 0 ↔ -𝐴 ≠ 0)
 
Theoremnegrebi 7264 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
𝐴 ∈ ℂ       (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)
 
Theoremnegne0i 7265 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       -𝐴 ≠ 0
 
Theoremsubcli 7266 Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴𝐵) ∈ ℂ
 
Theorempncan3i 7267 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + (𝐵𝐴)) = 𝐵
 
Theoremnegsubi 7268 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) = (𝐴𝐵)
 
Theoremsubnegi 7269 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 − -𝐵) = (𝐴 + 𝐵)
 
Theoremsubeq0i 7270 If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴𝐵) = 0 ↔ 𝐴 = 𝐵)
 
Theoremneg11i 7271 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (-𝐴 = -𝐵𝐴 = 𝐵)
 
Theoremnegcon1i 7272 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)
 
Theoremnegcon2i 7273 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 = -𝐵𝐵 = -𝐴)
 
Theoremnegdii 7274 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)
 
Theoremnegsubdii 7275 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       -(𝐴𝐵) = (-𝐴 + 𝐵)
 
Theoremnegsubdi2i 7276 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       -(𝐴𝐵) = (𝐵𝐴)
 
Theoremsubaddi 7277 Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴)
 
Theoremsubadd2i 7278 Relationship between subtraction and addition. (Contributed by NM, 15-Dec-2006.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴)
 
Theoremsubaddrii 7279 Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   (𝐵 + 𝐶) = 𝐴       (𝐴𝐵) = 𝐶
 
Theoremsubsub23i 7280 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = 𝐶 ↔ (𝐴𝐶) = 𝐵)
 
Theoremaddsubassi 7281 Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵𝐶))
 
Theoremaddsubi 7282 Law for subtraction and addition. (Contributed by NM, 6-Aug-2003.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) − 𝐶) = ((𝐴𝐶) + 𝐵)
 
Theoremsubcani 7283 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = (𝐴𝐶) ↔ 𝐵 = 𝐶)
 
Theoremsubcan2i 7284 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐶) = (𝐵𝐶) ↔ 𝐴 = 𝐵)
 
Theorempnncani 7285 Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) − (𝐴𝐶)) = (𝐵 + 𝐶)
 
Theoremaddsub4i 7286 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)) = ((𝐴𝐶) + (𝐵𝐷))
 
Theorem0reALT 7287 Alternate proof of 0re 7008. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℝ
 
Theoremnegcld 7288 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → -𝐴 ∈ ℂ)
 
Theoremsubidd 7289 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴𝐴) = 0)
 
Theoremsubid1d 7290 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 − 0) = 𝐴)
 
Theoremnegidd 7291 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 + -𝐴) = 0)
 
Theoremnegnegd 7292 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → --𝐴 = 𝐴)
 
Theoremnegeq0d 7293 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 = 0 ↔ -𝐴 = 0))
 
Theoremnegne0bd 7294 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 ≠ 0 ↔ -𝐴 ≠ 0))
 
Theoremnegcon1d 7295 Contraposition law for unary minus. Deduction form of negcon1 7242. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
 
Theoremnegcon1ad 7296 Contraposition law for unary minus. One-way deduction form of negcon1 7242. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → -𝐴 = 𝐵)       (𝜑 → -𝐵 = 𝐴)
 
Theoremneg11ad 7297 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 7241. Generalization of neg11d 7313. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 = -𝐵𝐴 = 𝐵))
 
Theoremnegned 7298 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 7313. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → -𝐴 ≠ -𝐵)
 
Theoremnegne0d 7299 The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → -𝐴 ≠ 0)
 
Theoremnegrebd 7300 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → -𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℝ)
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