mmtheorems55 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10688

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

Type	Label	Description
Statement
Theorem	negcon1t 5401	Negative contraposition law.
|- ((A e. CC /\ B e. CC) -> (-uA = B <-> -uB = A))
Theorem	negcon2t 5402	Negative contraposition law.
|- ((A e. CC /\ B e. CC) -> (A = -uB <-> B = -uA))
Theorem	subcant 5403	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) = (A - C) <-> B = C))
Theorem	subcan 5404	Cancellation law for subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - B) = (A - C) <-> B = C)
Theorem	subcan2 5405	Cancellation law for subtraction.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((A - C) = (B - C) <-> A = B)
Theorem	neg0 5406	Minus 0 equals 0.
|- -u0 = 0
Theorem	renegcl 5407	Closure law for negative of reals.
|- A e. RR   =>   |- -uA e. RR
Multiplication
Theorem	mulid2t 5408	Identity law for multiplication. Note: see ax1id 5273 for commuted version.
|- (A e. CC -> (1 x. A) = A)
Theorem	mul12t 5409	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 5410	Commutative/associative law.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. B) x. C) = ((A x. C) x. B))
Theorem	mul4t 5411	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 5412	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 5413	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 5414	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 5415	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 5416	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 5417	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 5418	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 5419	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 5420	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 5421	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 5422	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- A e. CC   =>   |- (A x. 0) = 0
Theorem	mul02 5423	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- A e. CC   =>   |- (0 x. A) = 0
Theorem	1p1times 5424	Two times a number.
|- A e. CC   =>   |- ((1 + 1) x. A) = (A + A)
Theorem	ine0 5425	The imaginary unit i is not zero.
|- i =/= 0
Theorem	1re 5426	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 5260, by exploiting properties of the imaginary unit i. (Contributed by Eric Schmidt, 11-Apr-2007.)
|- 1 e. RR
Theorem	peano2re 5427	A theorem for reals analogous the second Peano postulate peano2nn 5902.
|- (A e. RR -> (A + 1) e. RR)
Theorem	renegclt 5428	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 5407, to an antecedent.
|- (A e. RR -> -uA e. RR)
Theorem	resubclt 5429	Closure law for subtraction of reals.
|- ((A e. RR /\ B e. RR) -> (A - B) e. RR)
Theorem	resubcl 5430	Closure law for subtraction of reals.
|- A e. RR   &   |- B e. RR   =>   |- (A - B) e. RR
Theorem	0re 5431	0 is a real number. Proved without referencing 1re 5426. (Contributed by Eric Schmidt, 21-May-2007.)
|- 0 e. RR
Theorem	0reALT 5432	0 is a real number.
|- 0 e. RR
Theorem	peano2rem 5433	"Reverse" second Peano postulate analog for reals.
|- (N e. RR -> (N - 1) e. RR)
Theorem	mul01t 5434	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- (A e. CC -> (A x. 0) = 0)
Theorem	mul02t 5435	Multiplication by 0. Theorem I.6 of [Apostol] p. 18.
|- (A e. CC -> (0 x. A) = 0)
Theorem	mulneg1 5436	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 5437	Product with negative is negative of product.
|- A e. CC   &   |- B e. CC   =>   |- (A x. -uB) = -u(A x. B)
Theorem	mul2neg 5438	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 5439	Distribution of negative over addition.
|- A e. CC   &   |- B e. CC   =>   |- -u(A + B) = (-uA + -uB)
Theorem	negsubdi 5440	Distribution of negative over subtraction.
|- A e. CC   &   |- B e. CC   =>   |- -u(A - B) = (-uA + B)
Theorem	negsubdi2 5441	Distribution of negative over subtraction.
|- A e. CC   &   |- B e. CC   =>   |- -u(A - B) = (B - A)
Theorem	mulneg1t 5442	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 5443	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 5444	Swap the negative sign in a product.
|- ((A e. CC /\ B e. CC) -> (-uA x. B) = (A x. -uB))
Theorem	mul2negt 5445	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 5446	Distribution of negative over addition.
|- ((A e. CC /\ B e. CC) -> -u(A + B) = (-uA + -uB))
Theorem	negdi2t 5447	Distribution of negative over addition.
|- ((A e. CC /\ B e. CC) -> -u(A + B) = (-uA - B))
Theorem	negsubdit 5448	Distribution of negative over subtraction.
|- ((A e. CC /\ B e. CC) -> -u(A - B) = (-uA + B))
Theorem	negsubdi2t 5449	Distribution of negative over subtraction.
|- ((A e. CC /\ B e. CC) -> -u(A - B) = (B - A))
Theorem	neg2subt 5450	Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
|- ((A e. CC /\ B e. CC) -> (-uA - -uB) = (B - A))
Theorem	submul2t 5451	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 5452	Law for double subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A - (B - C)) = (A + (C - B)))
Theorem	subsubt 5453	Law for double subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A - (B - C)) = ((A - B) + C))
Theorem	subsub3t 5454	Law for double subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A - (B - C)) = ((A + C) - B))
Theorem	subsub4t 5455	Law for double subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - B) - C) = (A - (B + C)))
Theorem	sub23t 5456	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 5457	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A - (B - C)) - C) = (A - B))
Theorem	nnncan1t 5458	Cancellation law for subtraction.
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (