mmtheorems55 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5401-5500 - Page 55 of 108

Type	Label	Description
Statement
Theorem	pncan3t 5401	Subtraction and addition of equals.
|- ((A e. CC /\ B e. CC) -> (A + (B - A)) = B)
Theorem	pncan3 5402	Subtraction and addition of equals.
|- A e. CC   &   |- B e. CC   =>   |- (A + (B - A)) = B
Theorem	negidt 5403	Addition of a number and its negative.
|- (A e. CC -> (A + -uA) = 0)
Theorem	negid 5404	Addition of a number and its negative.
|- A e. CC   =>   |- (A + -uA) = 0
Theorem	negsub 5405	Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   =>   |- (A + -uB) = (A - B)
Theorem	negsubt 5406	Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18.
|- ((A e. CC /\ B e. CC) -> (A + -uB) = (A - B))
Theorem	addsubasst 5407	Associative-type law for addition and subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) - C) = (A + (B - C)))
Theorem	addsubt 5408	Law for addition and subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) - C) = ((A - C) + B))
Theorem	subadd23t 5409	Commutative/associative law for addition and subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) + C) = (A + (C - B)))
Theorem	addsub12t 5410	Commutative/associative law for addition and subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A + (B - C)) = (B + (A - C)))
Theorem	addsubass 5411	Associative-type law for subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) - C) = (A + (B - C))
Theorem	addsub 5412	Law for subtraction and addition.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A + B) - C) = ((A - C) + B)
Theorem	2addsubt 5413	Law for subtraction and addition.
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (((A + B) + C) - D) = (((A + C) - D) + B))
Theorem	negneg 5414	A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18.
|- A e. CC   =>   |- -u-uA = A
Theorem	subid 5415	Subtraction of a number from itself.
|- A e. CC   =>   |- (A - A) = 0
Theorem	subid1 5416	Identity law for subtraction.
|- A e. CC   =>   |- (A - 0) = A
Theorem	negnegt 5417	A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18.
|- (A e. CC -> -u-uA = A)
Theorem	subnegt 5418	Relationship between subtraction and negative.
|- ((A e. CC /\ B e. CC) -> (A - -uB) = (A + B))
Theorem	subidt 5419	Subtraction of a number from itself.
|- (A e. CC -> (A - A) = 0)
Theorem	subid1t 5420	Identity law for subtraction.
|- (A e. CC -> (A - 0) = A)
Theorem	pncant 5421	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC) -> ((A + B) - B) = A)
Theorem	pncan2t 5422	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC) -> ((A + B) - A) = B)
Theorem	npcant 5423	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC) -> ((A - B) + B) = A)
Theorem	npncant 5424	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) + (B - C)) = (A - C))
Theorem	nppcant 5425	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (((A - B) + C) + B) = (A + C))
Theorem	subcan2t 5426	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - C) = (B - C) <-> A = B))
Theorem	subeq0t 5427	If the difference between two numbers is zero, they are equal.
|- ((A e. CC /\ B e. CC) -> ((A - B) = 0 <-> A = B))
Theorem	subneg 5428	Relationship between subtraction and negative.
|- A e. CC   &   |- B e. CC   =>   |- (A - -uB) = (A + B)
Theorem	subeq0 5429	If the difference between two numbers is zero, they are equal.
|- A e. CC   &   |- B e. CC   =>   |- ((A - B) = 0 <-> A = B)
Theorem	neg11 5430	Negative is one-to-one.
|- A e. CC   &   |- B e. CC   =>   |- (-uA = -uB <-> A = B)
Theorem	negcon1 5431	Negative contraposition law.
|- A e. CC   &   |- B e. CC   =>   |- (-uA = B <-> -uB = A)
Theorem	negcon2 5432	Negative contraposition law.
|- A e. CC   &   |- B e. CC   =>   |- (A = -uB <-> B = -uA)
Theorem	neg11t 5433	Negative is one-to-one.
|- ((A e. CC /\ B e. CC) -> (-uA = -uB <-> A = B))
Theorem	negcon1t 5434	Negative contraposition law.
|- ((A e. CC /\ B e. CC) -> (-uA = B <-> -uB = A))
Theorem	negcon2t 5435	Negative contraposition law.
|- ((A e. CC /\ B e. CC) -> (A = -uB <-> B = -uA))
Theorem	subcant 5436	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) = (A - C) <-> B = C))
Theorem	subcan 5437	Cancellation law for subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = (A - C) <-> B = C)
Theorem	subcan2 5438	Cancellation law for subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - C) = (B - C) <-> A = B)
Theorem	neg0 5439	Minus 0 equals 0.
|- -u0 = 0
Theorem	renegcl 5440	Closure law for negative of reals.
|- A e. RR   =>   |- -uA e. RR
Multiplication
Theorem	mulid2t 5441	Identity law for multiplication. Note: see ax1id 5306 for commuted version.
|- (A e. CC -> (1 x. A) = A)
Theorem	mul12t 5442	Commutative/associative law for multiplication.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (B x. C)) = (B x. (A x. C)))
Theorem	mul23t 5443	Commutative/associative law.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. B) x. C) = ((A x. C) x. B))
Theorem	mul4t 5444	Rearrangement of 4 factors.
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A x. B) x. (C x. D)) = ((A x. C) x. (B x. D)))
Theorem	muladdt 5445	Product of two sums.
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + B) x. (C + D)) = (((A x. C) + (D x. B)) + ((A x. D) + (C x. B))))
Theorem	muladd11t 5446	A simple product of sums expansion.
|- ((A e. CC /\ B e. CC) -> ((1 + A) x. (1 + B)) = ((1 + A) + (B + (A x. B))))
Theorem	mul12 5447	Commutative/associative law that swaps the first two factors in a triple product.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (A x. (B x. C)) = (B x. (A x. C))
Theorem	mul23 5448	Commutative/associative law that swaps the last two factors in a triple product.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A x. B) x. C) = ((A x. C) x. B)
Theorem	mul4 5449	Rearrangement of 4 factors.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. CC   =>   |- ((A x. B) x. (C x. D)) = ((A x. C) x. (B x. D))
Theorem	muladd 5450	Product of two sums.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. CC   =>   |- ((A + B) x. (C + D)) = (((A x. C) + (D x. B)) + ((A x. D) + (C x. B)))
Theorem	subdit 5451	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (B - C)) = ((A x. B) - (A x. C)))
Theorem	subdirt 5452	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) x. C) = ((A x. C) - (B x. C)))
Theorem	subdi 5453	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (A x. (B - C)) = ((A x. B) - (A x. C))
Theorem	subdir 5454	Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) x. C) = ((A x. C) - (B x. C))
Theorem	mul01 5455	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- A e. CC   =>   |- (