mmtheorems60 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5901-6000 - Page 60 of 123

Type	Label	Description
Statement
Theorem	divcan3 5901	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> ((B x. A) / B) = A)
Theorem	divcan4 5902	A cancellation law for division.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> ((A x. B) / B) = A)
Theorem	div11i 5903	One-to-one relationship for division.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- C =/= 0   =>   |- ((A / C) = (B / C) <-> A = B)
Theorem	div11 5904	One-to-one relationship for division.
|- ((A e. CC /\ B e. CC /\ (C e. CC /\ C =/= 0)) -> ((A / C) = (B / C) <-> A = B))
Theorem	divid 5905	A number divided by itself is one.
|- ((A e. CC /\ A =/= 0) -> (A / A) = 1)
Theorem	div0 5906	Division into zero is zero.
|- ((A e. CC /\ A =/= 0) -> (0 / A) = 0)
Theorem	diveq0 5907	A ratio is zero iff the numerator is zero.
|- ((A e. CC /\ C e. CC /\ C =/= 0) -> ((A / C) = 0 <-> A = 0))
Theorem	recreci 5908	A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18.
|- A e. CC   &   |- A =/= 0   =>   |- (1 / (1 / A)) = A
Theorem	dividi 5909	A number divided by itself is one.
|- A e. CC   &   |- A =/= 0   =>   |- (A / A) = 1
Theorem	div0i 5910	Division into zero is zero.
|- A e. CC   &   |- A =/= 0   =>   |- (0 / A) = 0
Theorem	div1i 5911	A number divided by 1 is itself.
|- A e. CC   =>   |- (A / 1) = A
Theorem	div1 5912	A number divided by 1 is itself.
|- (A e. CC -> (A / 1) = A)
Theorem	divneg 5913	Move negative sign inside of a division.
|- ((A e. CC /\ B e. CC /\ B =/= 0) -> -u(A / B) = (-uA / B))
Theorem	divsubdir 5914	Distribution of division over subtraction.
|- ((A e. CC /\ B e. CC /\ (C e. CC /\ C =/= 0)) -> ((A - B) / C) = ((A / C) - (B / C)))
Theorem	recrec 5915	A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18.
|- ((A e. CC /\ A =/= 0) -> (1 / (1 / A)) = A)
Theorem	rec11ii 5916	Reciprocal is one-to-one.
|- A e. CC   &   |- B e. CC   &   |- A =/= 0   &   |- B =/= 0   =>   |- ((1 / A) = (1 / B) <-> A = B)
Theorem	rec11i 5917	Reciprocal is one-to-one.
|- A e. CC   &   |- B e. CC   =>   |- ((A =/= 0 /\ B =/= 0) -> ((1 / A) = (1 / B) <-> A = B))
Theorem	rec11r 5918	Mutual reciprocals. (Contributed by Paul Chapman, 18-Oct-2007.)
|- (((A e. CC /\ A =/= 0) /\ (B e. CC /\ B =/= 0)) -> ((1 / A) = B <-> (1 / B) = A))
Theorem	divmuldiv 5919	Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18.
|- (((A e. CC /\ B e. CC) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((A / C) x. (B / D)) = ((A x. B) / (C x. D)))
Theorem	divcan5 5920	Cancellation of common factor in a ratio.
|- ((A e. CC /\ (B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)) -> ((C x. A) / (C x. B)) = (A / B))
Theorem	divmul13 5921	Swap the denominators in the product of two ratios.
|- (((A e. CC /\ B e. CC) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((A / C) x. (B / D)) = ((B / C) x. (A / D)))
Theorem	divmul24 5922	Swap the numerators in the product of two ratios.
|- (((A e. CC /\ B e. CC) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((A / C) x. (B / D)) = ((A / D) x. (B / C)))
Theorem	divadddiv 5923	Addition of two ratios. Theorem I.13 of [Apostol] p. 18.
|- (((A e. CC /\ B e. CC) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((A / C) + (B / D)) = (((A x. D) + (C x. B)) / (C x. D)))
Theorem	divdivdiv 5924	Division of two ratios. Theorem I.15 of [Apostol] p. 18.
|- (((A e. CC /\ (B e. CC /\ B =/= 0)) /\ ((C e. CC /\ C =/= 0) /\ (D e. CC /\ D =/= 0))) -> ((A / B) / (C / D)) = ((A x. D) / (B x. C)))
Theorem	divmuldivi 5925	Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. CC   &   |- B =/= 0   &   |- D =/= 0   =>   |- ((A / B) x. (C / D)) = ((A x. C) / (B x. D))
Theorem	divmul13i 5926	Swap denominators of two ratios.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. CC   &   |- B =/= 0   &   |- D =/= 0   =>   |- ((A / B) x. (C / D)) = ((C / B) x. (A / D))
Theorem	divadddivi 5927	Addition of two ratios. Theorem I.13 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. CC   &   |- B =/= 0   &   |- D =/= 0   =>   |- ((A / B) + (C / D)) = (((A x. D) + (B x. C)) / (B x. D))
Theorem	divdivdivi 5928	Division of two ratios. Theorem I.15 of [Apostol] p. 18.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. CC   &   |- B =/= 0   &   |- D =/= 0   &   |- C =/= 0   =>   |- ((A / B) / (C / D)) = ((A x. D) / (B x. C))
Theorem	recdiv 5929	The reciprocal of a ratio.
|- (((A e. CC /\ A =/= 0) /\ (B e. CC /\ B =/= 0)) -> (1 / (A / B)) = (B / A))
Theorem	divcan6 5930	Cancellation of inverted fractions.
|- (((A e. CC /\ A =/= 0) /\ (B e. CC /\ B =/= 0)) -> ((A / B) x. (B / A)) = 1)
Theorem	divdiv23 5931	Swap denominators in a division.
|- ((A e. CC /\ (B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)) -> ((A / B) / C) = ((A / C) / B))
Theorem	divdiv23i 5932	Swap denominators in a division.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- B =/= 0   &   |- C =/= 0   =>   |- ((A / B) / C) = ((A / C) / B)
Theorem	divdiv23zi 5933	Swap denominators in a division.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- ((B =/= 0 /\ C =/= 0) -> ((A / B) / C) = ((A / C) / B))
Theorem	divdiv1 5934	Division into a fraction.
|- ((A e. CC /\ (B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)) -> ((A / B) / C) = (A / (B x. C)))
Theorem	divdiv2 5935	Division by a fraction.
|- ((A e. CC /\ (B e. CC /\ B =/= 0) /\ (C e. CC /\ C =/= 0)) -> (A / (B / C)) = ((A x. C) / B))
Theorem	recdiv2 5936	Division into a reciprocal.
|- (((A e. CC /\ A =/= 0) /\ (B e. CC /\ B =/= 0)) -> ((1 / A) / B) = (1 / (A x. B)))
Theorem	conjmul 5937	Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. Equation 5 of [Kreyszig] p. 12.
|- (((P e. CC /\ P =/= 0) /\ (Q e. CC /\ Q =/= 0)) -> (((1 / P) + (1 / Q)) = 1 <-> ((P - 1) x. (Q - 1)) = 1))
Theorem	redivcli 5938	Closure law for division of reals.
|- A e. RR   &   |- B e. RR   &   |- B =/= 0   =>   |- (A / B) e. RR
Theorem	redivclzi 5939	Closure law for division of reals.
|- A e. RR   &   |- B e. RR   =>   |- (B =/= 0 -> (A / B) e. RR)
Theorem	redivcl 5940	Closure law for division of reals.
|- ((A e. RR /\ B e. RR /\ B =/= 0) -> (A / B) e. RR)
Theorem	rereccli 5941	Closure law for reciprocal.
|- A e. RR   &   |- A =/= 0   =>   |- (1 / A) e. RR
Theorem	rerecclzi 5942	Closure law for reciprocal.
|- A e. RR   =>   |- (A =/= 0 -> (1 / A) e. RR)
Theorem	rereccl 5943	Closure law for reciprocal.
|- ((A e. RR /\ A =/= 0) -> (1 / A) e. RR)
Theorem	eqnegi 5944	A number equal to its negative is zero.
|- A e. CC   =>   |- (A = -uA <-> A = 0)
Theorem	eqneg 5945	A number equal to its negative is zero.
|- (A e. CC -> (A = -uA <-> A = 0))
Theorem	negeq0 5946	A number is zero iff its negative is zero.
|- (A e. CC -> (A = 0 <-> -uA = 0))
Theorem	negne0bi 5947	A number is non-zero iff its negative is non-zero.
|- A e. CC   =>   |- (A =/= 0 <-> -uA =/= 0)
Theorem	negne0i 5948	The negative of a non-zero number is non-zero.
|- A e. CC   &   |- A =/= 0   =>   |- -uA =/= 0
Ordering on reals (cont.)
Theorem	elimgt0 5949	Hypothesis for weak deduction theorem to eliminate 0 < A.
|- 0 < if(0 < A, A, 1)
Theorem	elimge0 5950	Hypothesis for weak deduction theorem to eliminate 0 <_ A.
|- 0 <_ if(0 <_ A, A, 0)
Theorem	ltp1 5951	A number is less than itself plus 1.
|- (A e. RR -> A < (A + 1))
Theorem	lep1 5952	A number is less than or equal to itself plus 1.
|- (A e. RR -> A <_ (A + 1))
Theorem	ltp1i 5953	A number is less than itself plus 1.
|- A e. RR   =>   |- A < (A + 1)
Theorem	recgt0ii 5954	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21.
|- A e. RR   &   |- 0 < A   =>   |- 0 < (1 / A)
Theorem	ltm1 5955	A number minus 1 is less than itself.
|- (A e. RR -> (A - 1) < A)
Theorem	letrp1 5956	A transitive property of 'less than or equal' and plus 1.
|- ((A e. RR /\ B e. RR /\ A <_ B) -> A <_ (B + 1))
Theorem	p1le 5957	A transitive property of plus 1 and 'less than or equal'.
|- ((A e. RR /\ B e. RR /\ (A + 1) <_ B) -> A <_ B)
Theorem	prodgt0lem 5958	Lemma for prodgt0i 5959.
Theorem	prodgt0i 5959	Infer that a multiplicand is positive from a nonnegative muliplier and positive product.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 < (A x. B)) -> 0 < B)
Theorem	prodge0i 5960	Infer that a multiplicand is nonnegative from a positive muliplier and nonnegative product.
|- A e. RR   &   |- B e. RR   =>   |- ((0 < A /\ 0 <_ (A x. B)) -> 0 <_ B)
Theorem	ltmul1ii 5961	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Proof shortened by Paul Chapman, 25-Jan-2008.)
|- A e. RR   &   |- B e. RR   &   |- C e. RR   &   |- 0 < C   =>   |- (A < B <-> (A x. C) < (B x. C))
Theorem	ltmul1i 5962	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- (0 < C -> (A < B <-> (A x. C) < (B x. C)))
Theorem	ltdiv1ii 5963	Division of both sides of 'less than' by a positive number.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   &   |- 0 < C   =>   |- (A < B <-> (A / C) < (B / C))
Theorem	ltdiv1i 5964	Division of both sides of 'less than' by a positive number.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- (0 < C -> (A < B <-> (A / C) < (B / C)))
Theorem	ltmuldivi 5965	'Less than' relationship between division and multiplication.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- (0 < C -> ((A x. C) < B <-> A < (B / C)))
Theorem	prodgt0 5966	Infer that a multiplicand is positive from a nonnegative muliplier and positive product.
|- (((A e. RR /\ B e. RR) /\ (0 <_ A /\ 0 < (A x. B))) -> 0 < B)
Theorem	prodgt02 5967	Infer that a multiplier is positive from a nonnegative muliplicand and positive product.
|- (((A e. RR /\ B e. RR) /\ (0 <_ B /\ 0 < (A x. B))) -> 0 < A)
Theorem	prodge0 5968	Infer that a multiplicand is nonnegative from a positive muliplier and nonnegative product.
|- (((A e. RR /\ B e. RR) /\ (0 < A /\ 0 <_ (A x. B))) -> 0 <_ B)
Theorem	prodge02 5969	Infer that a multiplier is nonnegative from a positive muliplicand and nonnegative product.
|- (((A e. RR /\ B e. RR) /\ (0 < B /\ 0 <_ (A x. B))) -> 0 <_ A)
Theorem	ltmul1 5970	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> (A < B <-> (A x. C) < (B x. C)))
Theorem	ltmul2 5971	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> (A < B <-> (C x. A) < (C x. B)))
Theorem	ltmul2OLD 5972	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A < B <-> (C x. A) < (C x. B)))
Theorem	lemul1 5973	Multiplication of both sides of 'less than or equal to' by a positive number.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> (A <_ B <-> (A x. C) <_ (B x. C)))
Theorem	lemul1OLD 5974	Multiplication of both sides of 'less than or equal to' by a positive number.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A <_ B <-> (A x. C) <_ (B x. C)))
Theorem	lemul2 5975	Multiplication of both sides of 'less than or equal to' by a positive number.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> (A <_ B <-> (C x. A) <_ (C x. B)))
Theorem	lemul2OLD 5976	Multiplication of both sides of 'less than or equal to' by a positive number.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A <_ B <-> (C x. A) <_ (C x. B)))
Theorem	ltmul2i 5977	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- (0 < C -> (A < B <-> (C x. A) < (C x. B)))
Theorem	ltmul2iOLD 5978	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- (0 < C -> (A < B <-> (C x. A) < (C x. B)))
Theorem	lemul1i 5979	Multiplication of both sides of 'less than or equal to' by a positive number.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- (0 < C -> (A <_ B <-> (A x. C) <_ (B x. C)))
Theorem	lemul2i 5980	Multiplication of both sides of 'less than or equal to' by a positive number.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- (0 < C -> (A <_ B <-> (C x. A) <_ (C x. B)))
Theorem	lemul1a 5981	Multiplication of both sides of 'less than or equal to' by a nonnegative number.
|- (((A e. RR /\ B e. RR /\ (C e. RR /\ 0 <_ C)) /\ A <_ B) -> (A x. C) <_ (B x. C))
Theorem	lemul1aOLD 5982	Multiplication of both sides of 'less than or equal to' by a nonnegative number.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ (0 <_ C /\ A <_ B)) -> (A x. C) <_ (B x. C))
Theorem	lemul2a 5983	Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
|- (((A e. RR /\ B e. RR /\ (C e. RR /\ 0 <_ C)) /\ A <_ B) -> (C x. A) <_ (C x. B))
Theorem	lemul2aOLD 5984	Multiplication of both sides of 'less than or equal to' by a nonnegative number.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ (0 <_ C /\ A <_ B)) -> (C x. A) <_ (C x. B))
Theorem	ltmul12a 5985	Comparison of product of two positive numbers.
|- ((((A e. RR /\ B e. RR) /\ (0 <_ A /\ A < B)) /\ ((C e. RR /\ D e. RR) /\ (0 <_ C /\ C < D))) -> (A x. C) < (B x. D))
Theorem	lemul12b 5986	Comparison of product of two nonnegative numbers.
|- ((((A e. RR /\ 0 <_ A) /\ B e. RR) /\ (C e. RR /\ (D e. RR /\ 0 <_ D))) -> ((A <_ B /\ C <_ D) -> (A x. C) <_ (B x. D)))
Theorem	lemul12aOLD 5987	Comparison of product of two nonnegative numbers.
|- ((((A e. RR /\ B e. RR) /\ (0 <_ A /\ A <_ B)) /\ ((C e. RR /\ D e. RR) /\ (0 <_ C /\ C <_ D))) -> (A x. C) <_ (B x. D))
Theorem	lemul12a 5988	Comparison of product of two nonnegative numbers.
|- ((((A e. RR /\ 0 <_ A) /\ B e. RR) /\ ((C e. RR /\ 0 <_ C) /\ D e. RR)) -> ((A <_ B /\ C <_ D) -> (A x. C) <_ (B x. D)))
Theorem	mulgt1 5989	The product of two numbers greater than 1 is greater than 1.
|- (((A e. RR /\ B e. RR) /\ (1 < A /\ 1 < B)) -> 1 < (A x. B))
Theorem	ltmulgt11 5990	Multiplication by a number greater than 1.
|- ((A e. RR /\ B e. RR /\ 0 < A) -> (1 < B <-> A < (A x. B)))
Theorem	ltmulgt12 5991	Multiplication by a number greater than 1.
|- ((A e. RR /\ B e. RR /\ 0 < A) -> (1 < B <-> A < (B x. A)))
Theorem	lemulge11 5992	Multiplication by a number greater than or equal to 1.
|- (((A e. RR /\ B e. RR) /\ (0 <_ A /\ 1 <_ B)) -> A <_ (A x. B))
Theorem	ltdiv1 5993	Division of both sides of 'less than' by a positive number.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> (A < B <-> (A / C) < (B / C)))
Theorem	ltdiv1OLD 5994	Division of both sides of 'less than' by a positive number.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A < B <-> (A / C) < (B / C)))
Theorem	lediv1 5995	Division of both sides of a less than or equal to relation by a positive number.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> (A <_ B <-> (A / C) <_ (B / C)))
Theorem	lediv1OLD 5996	Division of both sides of a less than or equal to relation by a positive number.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < C) -> (A <_ B <-> (A / C) <_ (B / C)))
Theorem	gt0div 5997	Division of a positive number by a positive number.
|- ((A e. RR /\ B e. RR /\ 0 < B) -> (0 < A <-> 0 < (A / B)))
Theorem	ge0div 5998	Division of a nonnegative number by a positive number.
|- ((A e. RR /\ B e. RR /\ 0 < B) -> (0 <_ A <-> 0 <_ (A / B)))
Theorem	divgt0 5999	The ratio of two positive numbers is positive.
|- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) -> 0 < (A / B))
Theorem	divge0 6000	The ratio of nonnegative and positive numbers is nonnegative.
|- (((A e. RR /\ 0 <_ A) /\ (B e. RR /\ 0 < B)) -> 0 <_ (A / B))