P. 115 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 11401-11500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pncan3 11401	Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
Theorem	npcan 11402	Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴)
Theorem	addsubass 11403	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 + (𝐵 − 𝐶)))
Theorem	addsub 11404	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) = ((𝐴 − 𝐶) + 𝐵))
Theorem	subadd23 11405	Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐶) = (𝐴 + (𝐶 − 𝐵)))
Theorem	addsub12 11406	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 11407	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 11408	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 11409	Subtraction and addition of equals. Almost but not exactly the same as pncan3i 11471 and pncan 11399, 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 11506. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) − 𝐵) = 𝐴
Theorem	mvrraddi 11410	Move the right term in a sum on the RHS to the LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐴 = (𝐵 + 𝐶)    ⇒   ⊢ (𝐴 − 𝐶) = 𝐵
Theorem	mvrladdi 11411	Move the left term in a sum on the RHS to the LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐴 = (𝐵 + 𝐶)    ⇒   ⊢ (𝐴 − 𝐵) = 𝐶
Theorem	mvlladdi 11412	Move the left term in a sum on the LHS to the RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 + 𝐵) = 𝐶    ⇒   ⊢ 𝐵 = (𝐶 − 𝐴)
Theorem	subid 11413	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 11414	Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)
Theorem	npncan 11415	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐵 − 𝐶)) = (𝐴 − 𝐶))
Theorem	nppcan 11416	Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 − 𝐵) + 𝐶) + 𝐵) = (𝐴 + 𝐶))
Theorem	nnpcan 11417	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 11418	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 11419	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵))
Theorem	subeq0 11420	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 11421	Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐵 − 𝐴)) = 0)
Theorem	subsub2 11422	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 11423	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 11424	Law for double subtraction. (Contributed by NM, 13-May-2004.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 − 𝐵) + 𝐶))
Theorem	nppcan2 11425	Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 + 𝐶)) + 𝐶) = (𝐴 − 𝐵))
Theorem	subsub3 11426	Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 + 𝐶) − 𝐵))
Theorem	subsub4 11427	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 11428	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 11429	Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 − 𝐶)) − 𝐶) = (𝐴 − 𝐵))
Theorem	nnncan1 11430	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 11431	Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶) − (𝐵 − 𝐶)) = (𝐴 − 𝐵))
Theorem	npncan3 11432	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 11433	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 11434	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 11435	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 11436	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 11437	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 11438	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 11439	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 11440	Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
⊢ -0 = 0
Theorem	negid 11441	Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0)
Theorem	negsub 11442	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 11443	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 11444	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 11445	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 11446	Negative contraposition law. (Contributed by NM, 9-May-2004.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
Theorem	negcon2 11447	Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = -𝐵 ↔ 𝐵 = -𝐴))
Theorem	negeq0 11448	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 11449	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 11450	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 11451	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 11452	Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 − 𝐵))
Theorem	negsubdi2 11453	Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (𝐵 − 𝐴))
Theorem	neg2sub 11454	Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 − -𝐵) = (𝐵 − 𝐴))
Theorem	renegcli 11455	Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 11457 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 11456	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 11457	Closure law for negative of reals. The weak deduction theorem dedth 4526 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 11455, to an antecedent. (Contributed by NM, 20-Jan-1997.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
Theorem	resubcl 11458	Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ)
Theorem	negreb 11459	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 11460	"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 11461	"Reverse" second Peano postulate analogue for reals. (Contributed by NM, 6-Feb-2007.)
⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)
Theorem	negcli 11462	Closure law for negative. (Contributed by NM, 26-Nov-1994.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ -𝐴 ∈ ℂ
Theorem	negidi 11463	Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 + -𝐴) = 0
Theorem	negnegi 11464	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 11465	Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 − 𝐴) = 0
Theorem	subid1i 11466	Identity law for subtraction. (Contributed by NM, 29-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 − 0) = 𝐴
Theorem	negne0bi 11467	A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 ≠ 0 ↔ -𝐴 ≠ 0)
Theorem	negrebi 11468	The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)
Theorem	negne0i 11469	The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    ⇒   ⊢ -𝐴 ≠ 0
Theorem	subcli 11470	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 11471	Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 + (𝐵 − 𝐴)) = 𝐵
Theorem	negsubi 11472	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 11473	Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 − -𝐵) = (𝐴 + 𝐵)
Theorem	subeq0i 11474	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 11475	Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵)
Theorem	negcon1i 11476	Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)
Theorem	negcon2i 11477	Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 = -𝐵 ↔ 𝐵 = -𝐴)
Theorem	negdii 11478	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 11479	Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ -(𝐴 − 𝐵) = (-𝐴 + 𝐵)
Theorem	negsubdi2i 11480	Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ -(𝐴 − 𝐵) = (𝐵 − 𝐴)
Theorem	subaddi 11481	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 11482	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 11483	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 11484	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 11485	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 11486	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 11487	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 − 𝐵) = (𝐴 − 𝐶) ↔ 𝐵 = 𝐶)
Theorem	subcan2i 11488	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵)
Theorem	pnncani 11489	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 11490	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 11491	Alternate proof of 0re 11146. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ∈ ℝ
Theorem	negcld 11492	Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → -𝐴 ∈ ℂ)
Theorem	subidd 11493	Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐴) = 0)
Theorem	subid1d 11494	Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 − 0) = 𝐴)
Theorem	negidd 11495	Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 + -𝐴) = 0)
Theorem	negnegd 11496	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 11497	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 11498	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 11499	Contraposition law for unary minus. Deduction form of negcon1 11446. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
Theorem	negcon1ad 11500	Contraposition law for unary minus. One-way deduction form of negcon1 11446. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → -𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → -𝐵 = 𝐴)