P. 116 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 11501-11600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mul31d 11501	Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))
Theorem	mul4d 11502	Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷)))
Theorem	muladd11r 11503	A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 1) · (𝐵 + 1)) = (((𝐴 · 𝐵) + (𝐴 + 𝐵)) + 1))
Theorem	comraddd 11504	Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐵))
Theorem	ltaddneg 11505	Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))
Theorem	ltaddnegr 11506	Adding a negative number to another number decreases it. (Contributed by AV, 19-Mar-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐴 + 𝐵) < 𝐵))
5.3  Real and complex numbers - basic operations
5.3.1  Addition
Theorem	add12 11507	Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))
Theorem	add32 11508	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))
Theorem	add32r 11509	Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 + 𝐶)) = ((𝐴 + 𝐶) + 𝐵))
Theorem	add4 11510	Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)))
Theorem	add42 11511	Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)))
Theorem	add12i 11512	Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))
Theorem	add32i 11513	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵)
Theorem	add4i 11514	Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐷 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))
Theorem	add42i 11515	Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ 𝐷 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))
Theorem	add12d 11516	Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))
Theorem	add32d 11517	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))
Theorem	add4d 11518	Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)))
Theorem	add42d 11519	Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)))
5.3.2  Subtraction
Syntax	cmin 11520	Extend class notation to include subtraction.
class −
Syntax	cneg 11521	Extend class notation to include unary minus. The symbol - is not a class by itself but part of a compound class definition. We do this rather than making it a formal function since it is so commonly used. Note: We use different symbols for unary minus (-) and subtraction cmin 11520 (−) to prevent syntax ambiguity. For example, looking at the syntax definition co 7448, if we used the same symbol then "( − 𝐴 − 𝐵) " could mean either "− 𝐴 " minus "𝐵", or it could represent the (meaningless) operation of classes "− " and "− 𝐵 " connected with "operation" "𝐴". On the other hand, "(-𝐴 − 𝐵) " is unambiguous.
class -𝐴
Definition	df-sub 11522*	Define subtraction. Theorem subval 11527 shows its value (and describes how this definition works), Theorem subaddi 11623 relates it to addition, and Theorems subcli 11612 and resubcli 11598 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
Definition	df-neg 11523	Define the negative of a number (unary minus). We use different symbols for unary minus (-) and subtraction (−) to prevent syntax ambiguity. See cneg 11521 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
⊢ -𝐴 = (0 − 𝐴)
Theorem	0cnALT 11524	Alternate proof of 0cn 11282 which does not reference ax-1cn 11242. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ∈ ℂ
Theorem	0cnALT2 11525	Alternate proof of 0cnALT 11524 which is shorter, but depends on ax-8 2110, ax-13 2380, ax-sep 5317, ax-nul 5324, ax-pow 5383, ax-pr 5447, ax-un 7770, and every complex number axiom except ax-pre-mulgt0 11261 and ax-pre-sup 11262. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ∈ ℂ
Theorem	negeu 11526*	Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
Theorem	subval 11527*	Value of subtraction, which is the (unique) element 𝑥 such that 𝐵 + 𝑥 = 𝐴. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
Theorem	negeq 11528	Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵)
Theorem	negeqi 11529	Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ -𝐴 = -𝐵
Theorem	negeqd 11530	Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → -𝐴 = -𝐵)
Theorem	nfnegd 11531	Deduction version of nfneg 11532. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥-𝐴)
Theorem	nfneg 11532	Bound-variable hypothesis builder for the negative of a complex number. (Contributed by NM, 12-Jun-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥-𝐴
Theorem	csbnegg 11533	Move class substitution in and out of the negative of a number. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌-𝐵 = -⦋𝐴 / 𝑥⦌𝐵)
Theorem	negex 11534	A negative is a set. (Contributed by NM, 4-Apr-2005.)
⊢ -𝐴 ∈ V
Theorem	subcl 11535	Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ)
Theorem	negcl 11536	Closure law for negative. (Contributed by NM, 6-Aug-2003.)
⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
Theorem	negicn 11537	-i is a complex number. (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ -i ∈ ℂ
Theorem	subf 11538	Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
⊢ − :(ℂ × ℂ)⟶ℂ
Theorem	subadd 11539	Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
Theorem	subadd2 11540	Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴))
Theorem	subsub23 11541	Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐴 − 𝐶) = 𝐵))
Theorem	pncan 11542	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	pncan2 11543	Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵)
Theorem	pncan3 11544	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 11545	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 11546	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 11547	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 11548	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 11549	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 11550	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 11551	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 11552	Subtraction and addition of equals. Almost but not exactly the same as pncan3i 11613 and pncan 11542, 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 11648. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) − 𝐵) = 𝐴
Theorem	mvrraddi 11553	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	mvlladdi 11554	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 11555	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 11556	Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)
Theorem	npncan 11557	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐵 − 𝐶)) = (𝐴 − 𝐶))
Theorem	nppcan 11558	Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴 − 𝐵) + 𝐶) + 𝐵) = (𝐴 + 𝐶))
Theorem	nnpcan 11559	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 11560	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 11561	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵))
Theorem	subeq0 11562	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 11563	Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + (𝐵 − 𝐴)) = 0)
Theorem	subsub2 11564	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 11565	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 11566	Law for double subtraction. (Contributed by NM, 13-May-2004.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 − 𝐵) + 𝐶))
Theorem	nppcan2 11567	Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 + 𝐶)) + 𝐶) = (𝐴 − 𝐵))
Theorem	subsub3 11568	Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 + 𝐶) − 𝐵))
Theorem	subsub4 11569	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 11570	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 11571	Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 − 𝐶)) − 𝐶) = (𝐴 − 𝐵))
Theorem	nnncan1 11572	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 11573	Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶) − (𝐵 − 𝐶)) = (𝐴 − 𝐵))
Theorem	npncan3 11574	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 11575	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 11576	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 11577	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 11578	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 11579	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 11580	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 11581	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 11582	Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
⊢ -0 = 0
Theorem	negid 11583	Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0)
Theorem	negsub 11584	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 11585	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 11586	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 11587	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 11588	Negative contraposition law. (Contributed by NM, 9-May-2004.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
Theorem	negcon2 11589	Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = -𝐵 ↔ 𝐵 = -𝐴))
Theorem	negeq0 11590	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 11591	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 11592	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 11593	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 11594	Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 − 𝐵))
Theorem	negsubdi2 11595	Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (𝐵 − 𝐴))
Theorem	neg2sub 11596	Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 − -𝐵) = (𝐵 − 𝐴))
Theorem	renegcli 11597	Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 11599 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 11598	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 11599	Closure law for negative of reals. The weak deduction theorem dedth 4606 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 11597, to an antecedent. (Contributed by NM, 20-Jan-1997.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
Theorem	resubcl 11600	Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ)