mmtheorems55 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5401-5500 - Page 55 of 107

Type	Label	Description
Statement
Theorem	negcon2t 5401	Negative contraposition law.
|- ((A e. CC /\ B e. CC) -> (A = -uB <-> B = -uA))
Theorem	subcant 5402	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) = (A - C) <-> B = C))
Theorem	subcan 5403	Cancellation law for subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = (A - C) <-> B = C)
Theorem	subcan2 5404	Cancellation law for subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - C) = (B - C) <-> A = B)
Theorem	neg0 5405	Minus 0 equals 0.
|- -u0 = 0
Theorem	renegcl 5406	Closure law for negative of reals.
|- A e. RR   =>   |- -uA e. RR
Multiplication
Theorem	mulid2t 5407	Identity law for multiplication. Note: see ax1id 5272 for commuted version.
|- (A e. CC -> (1 x. A) = A)
Theorem	mul12t 5408	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 5409	Commutative/associative law.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. B) x. C) = ((A x. C) x. B))
Theorem	mul4t 5410	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 5411	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 5412	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 5413	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 5414	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 5415	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 5416	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 5417	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 5418	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 5419	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 5420	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 5421	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- A e. CC   =>   |- (A x. 0) = 0
Theorem	mul02 5422	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- A e. CC   =>   |- (0 x. A) = 0
Theorem	1p1times 5423	Two times a number.
|- A e. CC   =>   |- ((1 + 1) x. A) = (A + A)
Theorem	ine0 5424	The imaginary unit i is not zero.
|- i =/= 0
Theorem	1re 5425	1 is a real number. This used to be one of our postulates for complex numbers, but Eric Schmidt discovered that it could be derived from a weaker postulate, ax1cn 5259, by exploiting properties of the imaginary unit i. (Contributed by Eric Schmidt, 11-Apr-2007.)
|- 1 e. RR
Theorem	peano2re 5426	A theorem for reals analogous the second Peano postulate peano2nn 5901.
|- (A e. RR -> (A + 1) e. RR)
Theorem	renegclt 5427	Closure law for negative of reals. The weak deduction theorem dedth 2379 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcl 5406, to an antecedent.
|- (A e. RR -> -uA e. RR)
Theorem	resubclt 5428	Closure law for subtraction of reals.
|- ((A e. RR /\ B e. RR) -> (A - B) e. RR)
Theorem	resubcl 5429	Closure law for subtraction of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A - B) e. RR
Theorem	0re 5430	0 is a real number. Proved without referencing 1re 5425. (Contributed by Eric Schmidt, 21-May-2007.)
|- 0 e. RR
Theorem	0reALT 5431	0 is a real number.
|- 0 e. RR
Theorem	peano2rem 5432	"Reverse" second Peano postulate analog for reals.
|- (N e. RR -> (N - 1) e. RR)
Theorem	mul01t 5433	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- (A e. CC -> (A x. 0) = 0)
Theorem	mul02t 5434	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- (A e. CC -> (0 x. A) = 0)
Theorem	mulneg1 5435	Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   =>   |- (-uA x. B) = -u(A x. B)
Theorem	mulneg2 5436	Product with negative is negative of product.
|- A e. CC   &   |- B e. CC   =>   |- (A x. -uB) = -u(A x. B)
Theorem	mul2neg 5437	Product of two negatives. Theorem I.12 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   =>   |- (-uA x. -uB) = (A x. B)
Theorem	negdi 5438	Distribution of negative over addition.
|- A e. CC   &   |- B e. CC   =>   |- -u(A + B) = (-uA + -uB)
Theorem	negsubdi 5439	Distribution of negative over subtraction.
|- A e. CC   &   |- B e. CC   =>   |- -u(A - B) = (-uA + B)
Theorem	negsubdi2 5440	Distribution of negative over subtraction.
|- A e. CC   &   |- B e. CC   =>   |- -u(A - B) = (B - A)
Theorem	mulneg1t 5441	Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18.
|- ((A e. CC /\ B e. CC) -> (-uA x. B) = -u(A x. B))
Theorem	mulneg2t 5442	The product with a negative is the negative of the product.
|- ((A e. CC /\ B e. CC) -> (A x. -uB) = -u(A x. B))
Theorem	mulneg12t 5443	Swap the negative sign in a product.
|- ((A e. CC /\ B e. CC) -> (-uA x. B) = (A x. -uB))
Theorem	mul2negt 5444	Product of two negatives. Theorem I.12 of [Apostol] p. 18.
|- ((A e. CC /\ B e. CC) -> (-uA x. -uB) = (A x. B))
Theorem	negdit 5445	Distribution of negative over addition.
|- ((A e. CC /\ B e. CC) -> -u(A + B) = (-uA + -uB))
Theorem	negdi2t 5446	Distribution of negative over addition.
|- ((A e. CC /\ B e. CC) -> -u(A + B) = (-uA - B))
Theorem	negsubdit 5447	Distribution of negative over subtraction.
|- ((A e. CC /\ B e. CC) -> -u(A - B) = (-uA + B))
Theorem	negsubdi2t 5448	Distribution of negative over subtraction.
|- ((A e. CC /\ B e. CC) -> -u(A - B) = (B - A))
Theorem	neg2subt 5449	Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
|- ((A e. CC /\ B e. CC) -> (-uA - -uB) = (B - A))
Theorem	submul2t 5450	Convert a subtraction to addition using multiplication by a negative.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A - (B x. C)) = (A + (B x. -uC)))
Theorem	subsub2t 5451	Law for double subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A - (B - C)) = (A + (C - B)))
Theorem	subsubt 5452	Law for double subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A - (B - C)) = ((A - B) + C))
Theorem	subsub3t 5453	Law for double subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A - (B - C)) = ((A + C) - B))
Theorem	subsub4t 5454	Law for double subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) - C) = (A - (B + C)))
Theorem	sub23t 5455	Swap the second and third terms in a double subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) - C) = ((A - C) - B))
Theorem	nnncant 5456	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - (B - C)) - C) = (A - B))
Theorem	nnncan1t 5457	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) - (A - C)) = (C - B))
Theorem	nnncan2t 5458	Cancellation law for subtraction.
|- ((A e.