mmtheorems59 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5801-5900 - Page 59 of 107

Type	Label	Description
Statement
Theorem	ltmul2 5801	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 5802	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 5803	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	lemul1it 5804	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	lemul1itOLD 5805	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	lemul2it 5806	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	lemul2itOLD 5807	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	ltmul12it 5808	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	lemul12ait 5809	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	lemul12itOLD 5810	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	lemul12it 5811	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	mulgt1t 5812	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	ltmulgt11t 5813	Multiplication by a number greater than 1.
|- ((A e. RR /\ B e. RR /\ 0 < A) -> (1 < B <-> A < (A x. B)))
Theorem	ltmulgt12t 5814	Multiplication by a number greater than 1.
|- ((A e. RR /\ B e. RR /\ 0 < A) -> (1 < B <-> A < (B x. A)))
Theorem	lemulge11t 5815	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	ltdiv1t 5816	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	lediv1t 5817	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	gt0divt 5818	Division of a positive number by a positive number.
|- ((A e. RR /\ B e. RR /\ 0 < B) -> (0 < A <-> 0 < (A / B)))
Theorem	ge0divt 5819	Division of a nonnegative number by a positive number.
|- ((A e. RR /\ B e. RR /\ 0 < B) -> (0 <_ A <-> 0 <_ (A / B)))
Theorem	divgt0t 5820	The ratio of two positive numbers is positive.
|- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) -> 0 < (A / B))
Theorem	divge0t 5821	The ratio of nonnegative and positive numbers is nonnegative.
|- (((A e. RR /\ 0 <_ A) /\ (B e. RR /\ 0 < B)) -> 0 <_ (A / B))
Theorem	divgt0 5822	The ratio of two positive numbers is positive.
|- A e. RR   &   |- B e. RR   =>   |- ((0 < A /\ 0 < B) -> 0 < (A / B))
Theorem	divge0 5823	The ratio of nonnegative and positive numbers is nonnegative.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 < B) -> 0 <_ (A / B))
Theorem	divgt0i2 5824	The ratio of two positive numbers is positive.
|- A e. RR   &   |- B e. RR   &   |- 0 < B   =>   |- (0 < A -> 0 < (A / B))
Theorem	divgt0i 5825	The ratio of two positive numbers is positive.
|- A e. RR   &   |- B e. RR   &   |- 0 < A   &   |- 0 < B   =>   |- 0 < (A / B)
Theorem	recgt0t 5826	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21.
|- ((A e. RR /\ 0 < A) -> 0 < (1 / A))
Theorem	recgt0 5827	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21.
|- A e. RR   =>   |- (0 < A -> 0 < (1 / A))
Theorem	ltmuldivt 5828	'Less than' relationship between division and multiplication.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < B) -> ((A x. B) < C <-> A < (C / B)))
Theorem	ltmuldiv2t 5829	'Less than' relationship between division and multiplication.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < A) -> ((A x. B) < C <-> B < (C / A)))
Theorem	ltdivmult 5830	'Less than' relationship between division and multiplication.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < B) -> ((A / B) < C <-> A < (B x. C)))
Theorem	ledivmult 5831	'Less than or equal to' relationship between division and multiplication.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < B) -> ((A / B) <_ C <-> A <_ (B x. C)))
Theorem	ltdivmul2t 5832	'Less than' relationship between division and multiplication.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < B) -> ((A / B) < C <-> A < (C x. B)))
Theorem	lt2mul2divt 5833	'Less than' relationship between division and multiplication.
|- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR) /\ (0 < B /\ 0 < D)) -> ((A x. B) < (C x. D) <-> (A / D) < (C / B)))
Theorem	ledivmul2t 5834	'Less than or equal to' relationship between division and multiplication.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < B) -> ((A / B) <_ C <-> A <_ (C x. B)))
Theorem	lemuldivt 5835	'Less than or equal' relationship between division and multiplication.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < B) -> ((A x. B) <_ C <-> A <_ (C / B)))
Theorem	lemuldiv2t 5836	'Less than or equal' relationship between division and multiplication.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ 0 < A) -> ((A x. B) <_ C <-> B <_ (C / A)))
Theorem	ltreci 5837	The reciprocal of both sides of 'less than'.
|- A e. RR   &   |- B e. RR   &   |- 0 < A   &   |- 0 < B   =>   |- (A < B <-> (1 / B) < (1 / A))
Theorem	ltrec 5838	The reciprocal of both sides of 'less than'.
|- A e. RR   &   |- B e. RR   =>   |- ((0 < A /\ 0 < B) -> (A < B <-> (1 / B) < (1 / A)))
Theorem	lerec 5839	The reciprocal of both sides of 'less than or equal to'.
|- A e. RR   &   |- B e. RR   =>   |- ((0 < A /\ 0 < B) -> (A <_ B <-> (1 / B) <_ (1 / A)))
Theorem	lt2msq 5840	The square function on nonnegative reals is strictly monotonic.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> (A < B <-> (A x. A) < (B x. B)))
Theorem	le2msq 5841	The square function on nonnegative reals is monotonic.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> (A <_ B <-> (A x. A) <_ (B x. B)))
Theorem	msq11 5842	The square of a nonnegative number is a one-to-one function.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 <_ B) -> ((A x. A) = (B x. B) <-> A = B))
Theorem	ltrect 5843	The reciprocal of both sides of 'less than'.
|- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) -> (A < B <-> (1 / B) < (1 / A)))
Theorem	lerect 5844	The reciprocal of both sides of 'less than or equal to'.
|- ((