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	mul12t 5401	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 5402	Commutative/associative law.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. B) x. C) = ((A x. C) x. B))
Theorem	mul4t 5403	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 5404	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 5405	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 5406	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 5407	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 5408	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 5409	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 5410	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 5411	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 5412	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 5413	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 5414	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- A e. CC   =>   |- (A x. 0) = 0
Theorem	mul02 5415	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- A e. CC   =>   |- (0 x. A) = 0
Theorem	1p1times 5416	Two times a number.
|- A e. CC   =>   |- ((1 + 1) x. A) = (A + A)
Theorem	ine0 5417	The imaginary unit i is not zero.
|- i =/= 0
Theorem	1re 5418	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 5252, by exploiting properties of the imaginary unit i. (Contributed by Eric Schmidt, 11-Apr-2007.)
|- 1 e. RR
Theorem	peano2re 5419	A theorem for reals analogous the second Peano postulate peano2nn 5894.
|- (A e. RR -> (A + 1) e. RR)
Theorem	renegclt 5420	Closure law for negative of reals. The weak deduction theorem dedth 2380 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcl 5399, to an antecedent.
|- (A e. RR -> -uA e. RR)
Theorem	resubclt 5421	Closure law for subtraction of reals.
|- ((A e. RR /\ B e. RR) -> (A - B) e. RR)
Theorem	resubcl 5422	Closure law for subtraction of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A - B) e. RR
Theorem	0re 5423	0 is a real number. Proved without referencing 1re 5418. (Contributed by Eric Schmidt, 21-May-2007.)
|- 0 e. RR
Theorem	0reALT 5424	0 is a real number.
|- 0 e. RR
Theorem	peano2rem 5425	"Reverse" second Peano postulate analog for reals.
|- (N e. RR -> (N - 1) e. RR)
Theorem	mul01t 5426	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- (A e. CC -> (A x. 0) = 0)
Theorem	mul02t 5427	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- (A e. CC -> (0 x. A) = 0)
Theorem	mulneg1 5428	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 5429	Product with negative is negative of product.
|- A e. CC   &   |- B e. CC   =>   |- (A x. -uB) = -u(A x. B)
Theorem	mul2neg 5430	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 5431	Distribution of negative over addition.
|- A e. CC   &   |- B e. CC   =>   |- -u(A + B) = (-uA + -uB)
Theorem	negsubdi 5432	Distribution of negative over subtraction.
|- A e. CC   &   |- B e. CC   =>   |- -u(A - B) = (-uA + B)
Theorem	negsubdi2 5433	Distribution of negative over subtraction.
|- A e. CC   &   |- B e. CC   =>   |- -u(A - B) = (B - A)
Theorem	mulneg1t 5434	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 5435	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 5436	Swap the negative sign in a product.
|- ((A e. CC /\ B e. CC) -> (-uA x. B) = (A x. -uB))
Theorem	mul2negt 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	negdit 5438	Distribution of negative over addition.
|- ((A e. CC /\ B e. CC) -> -u(A + B) = (-uA + -uB))
Theorem	negdi2t 5439	Distribution of negative over addition.
|- ((A e. CC /\ B e. CC) -> -u(A + B) = (-uA - B))
Theorem	negsubdit 5440	Distribution of negative over subtraction.
|- ((A e. CC /\ B e. CC) -> -u(A - B) = (-uA + B))
Theorem	negsubdi2t 5441	Distribution of negative over subtraction.
|- ((A e. CC /\ B e. CC) -> -u(A - B) = (B - A))
Theorem	neg2subt 5442	Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
|- ((A e. CC /\ B e. CC) -> (-uA - -uB) = (B - A))
Theorem	submul2t 5443	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 5444	Law for double subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A - (B - C)) = (A + (C - B)))
Theorem	subsubt 5445	Law for double subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A - (B - C)) = ((A - B) + C))
Theorem	subsub3t 5446	Law for double subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A - (B - C)) = ((A + C) - B))
Theorem	subsub4t 5447	Law for double subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) - C) = (A - (B + C)))
Theorem	sub23t 5448	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 5449	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - (B - C)) - C) = (A - B))
Theorem	nnncan1t 5450	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) - (A - C)) = (C - B))
Theorem	nnncan2t 5451	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - C) - (B - C)) = (A - B))
Theorem	nncant 5452	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC) -> (A - (A - B)) = B)
Theorem	nppcan2t 5453	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - (B + C)) + C) = (A - B))
Theorem	mulm1t 5454	Product with minus one is negative.
|- (A e. CC -> (-u1 x. A) = -uA)
Theorem	mulm1 5455	Product with minus one is negative.
|- A e. CC   =>   |- (-u1 x. A) = -uA
Theorem	addsub4t 5456	Rearrangement of 4 terms in a mixed addition and subtraction.
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + B) - (C + D)) = ((A - C) + (B - D)))
Theorem