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	negsub 11401	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 11402	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 11403	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 11404	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 11405	Negative contraposition law. (Contributed by NM, 9-May-2004.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
Theorem	negcon2 11406	Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = -𝐵 ↔ 𝐵 = -𝐴))
Theorem	negeq0 11407	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 11408	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 11409	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 11410	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 11411	Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 − 𝐵))
Theorem	negsubdi2 11412	Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (𝐵 − 𝐴))
Theorem	neg2sub 11413	Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 − -𝐵) = (𝐵 − 𝐴))
Theorem	renegcli 11414	Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 11416 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 11415	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 11416	Closure law for negative of reals. The weak deduction theorem dedth 4532 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 11414, to an antecedent. (Contributed by NM, 20-Jan-1997.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
Theorem	resubcl 11417	Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ)
Theorem	negreb 11418	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 11419	"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 11420	"Reverse" second Peano postulate analogue for reals. (Contributed by NM, 6-Feb-2007.)
⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)
Theorem	negcli 11421	Closure law for negative. (Contributed by NM, 26-Nov-1994.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ -𝐴 ∈ ℂ
Theorem	negidi 11422	Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 + -𝐴) = 0
Theorem	negnegi 11423	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 11424	Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 − 𝐴) = 0
Theorem	subid1i 11425	Identity law for subtraction. (Contributed by NM, 29-May-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 − 0) = 𝐴
Theorem	negne0bi 11426	A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 ≠ 0 ↔ -𝐴 ≠ 0)
Theorem	negrebi 11427	The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)
Theorem	negne0i 11428	The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    ⇒   ⊢ -𝐴 ≠ 0
Theorem	subcli 11429	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 11430	Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 + (𝐵 − 𝐴)) = 𝐵
Theorem	negsubi 11431	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 11432	Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 − -𝐵) = (𝐴 + 𝐵)
Theorem	subeq0i 11433	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 11434	Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵)
Theorem	negcon1i 11435	Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)
Theorem	negcon2i 11436	Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 = -𝐵 ↔ 𝐵 = -𝐴)
Theorem	negdii 11437	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 11438	Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ -(𝐴 − 𝐵) = (-𝐴 + 𝐵)
Theorem	negsubdi2i 11439	Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ -(𝐴 − 𝐵) = (𝐵 − 𝐴)
Theorem	subaddi 11440	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 11441	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 11442	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 11443	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 11444	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 11445	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 11446	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 − 𝐵) = (𝐴 − 𝐶) ↔ 𝐵 = 𝐶)
Theorem	subcan2i 11447	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵)
Theorem	pnncani 11448	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 11449	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 11450	Alternate proof of 0re 11106. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ∈ ℝ
Theorem	negcld 11451	Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → -𝐴 ∈ ℂ)
Theorem	subidd 11452	Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐴) = 0)
Theorem	subid1d 11453	Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 − 0) = 𝐴)
Theorem	negidd 11454	Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 + -𝐴) = 0)
Theorem	negnegd 11455	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 11456	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 11457	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 11458	Contraposition law for unary minus. Deduction form of negcon1 11405. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
Theorem	negcon1ad 11459	Contraposition law for unary minus. One-way deduction form of negcon1 11405. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → -𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → -𝐵 = 𝐴)
Theorem	neg11ad 11460	The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 11404. Generalization of neg11d 11476. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵))
Theorem	negned 11461	If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 11476. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → -𝐴 ≠ -𝐵)
Theorem	negne0d 11462	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 11463	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 11464	Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ)
Theorem	pncand 11465	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴)
Theorem	pncan2d 11466	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐴) = 𝐵)
Theorem	pncan3d 11467	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 11468	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐵) = 𝐴)
Theorem	nncand 11469	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) = 𝐵)
Theorem	negsubd 11470	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 11471	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 11472	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 11473	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)
Theorem	subeq0ad 11474	The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 11379. Generalization of subeq0d 11472. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵))
Theorem	subne0ad 11475	If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 11473. Contrapositive of subeq0bd 11535. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	neg11d 11476	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)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	negdid 11477	Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
Theorem	negdi2d 11478	Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → -(𝐴 + 𝐵) = (-𝐴 − 𝐵))
Theorem	negsubdid 11479	Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → -(𝐴 − 𝐵) = (-𝐴 + 𝐵))
Theorem	negsubdi2d 11480	Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → -(𝐴 − 𝐵) = (𝐵 − 𝐴))
Theorem	neg2subd 11481	Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (-𝐴 − -𝐵) = (𝐵 − 𝐴))
Theorem	subaddd 11482	Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
Theorem	subadd2d 11483	Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴))
Theorem	addsubassd 11484	Associative-type law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵 − 𝐶)))
Theorem	addsubd 11485	Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵))
Theorem	subadd23d 11486	Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) + 𝐶) = (𝐴 + (𝐶 − 𝐵)))
Theorem	addsub12d 11487	Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐶)) = (𝐵 + (𝐴 − 𝐶)))
Theorem	npncand 11488	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) + (𝐵 − 𝐶)) = (𝐴 − 𝐶))
Theorem	nppcand 11489	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (((𝐴 − 𝐵) + 𝐶) + 𝐵) = (𝐴 + 𝐶))
Theorem	nppcan2d 11490	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − (𝐵 + 𝐶)) + 𝐶) = (𝐴 − 𝐵))
Theorem	nppcan3d 11491	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) + (𝐶 + 𝐵)) = (𝐴 + 𝐶))
Theorem	subsubd 11492	Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 − 𝐵) + 𝐶))
Theorem	subsub2d 11493	Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 − (𝐵 − 𝐶)) = (𝐴 + (𝐶 − 𝐵)))
Theorem	subsub3d 11494	Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 − (𝐵 − 𝐶)) = ((𝐴 + 𝐶) − 𝐵))
Theorem	subsub4d 11495	Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) − 𝐶) = (𝐴 − (𝐵 + 𝐶)))
Theorem	sub32d 11496	Swap the second and third terms in a double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) − 𝐶) = ((𝐴 − 𝐶) − 𝐵))
Theorem	nnncand 11497	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − (𝐵 − 𝐶)) − 𝐶) = (𝐴 − 𝐵))
Theorem	nnncan1d 11498	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) − (𝐴 − 𝐶)) = (𝐶 − 𝐵))
Theorem	nnncan2d 11499	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐶) − (𝐵 − 𝐶)) = (𝐴 − 𝐵))
Theorem	npncan3d 11500	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) + (𝐶 − 𝐴)) = (𝐶 − 𝐵))