mmtheorems59 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5801-5900 - Page 59 of 108

Type	Label	Description
Statement
Theorem	redivclz 5801	Closure law for division of reals.
|- A e. RR   &   |- B e. RR   =>   |- (B =/= 0 -> (A / B) e. RR)
Theorem	redivclt 5802	Closure law for division of reals.
|- ((A e. RR /\ B e. RR /\ B =/= 0) -> (A / B) e. RR)
Theorem	rereccl 5803	Closure law for reciprocal.
|- A e. RR   &   |- A =/= 0   =>   |- (1 / A) e. RR
Theorem	rerecclz 5804	Closure law for reciprocal.
|- A e. RR   =>   |- (A =/= 0 -> (1 / A) e. RR)
Theorem	rerecclt 5805	Closure law for reciprocal.
|- ((A e. RR /\ A =/= 0) -> (1 / A) e. RR)
Theorem	eqneg 5806	A number equal to its negative is zero.
|- A e. CC   =>   |- (A = -uA <-> A = 0)
Theorem	eqnegt 5807	A number equal to its negative is zero.
|- (A e. CC -> (A = -uA <-> A = 0))
Theorem	negeq0t 5808	A number is zero iff its negative is zero.
|- (A e. CC -> (A = 0 <-> -uA = 0))
Theorem	negne0 5809	A number is non-zero iff its negative is non-zero.
|- A e. CC   =>   |- (A =/= 0 <-> -uA =/= 0)
Theorem	negn0 5810	The negative of a non-zero number is non-zero.
|- A e. CC   &   |- A =/= 0   =>   |- -uA =/= 0
Ordering on reals (cont.)
Theorem	elimgt0 5811	Hypothesis for weak deduction theorem to eliminate 0 < A.
|- 0 < if(0 < A, A, 1)
Theorem	elimge0 5812	Hypothesis for weak deduction theorem to eliminate 0 <_ A.
|- 0 <_ if(0 <_ A, A, 0)
Theorem	ltp1t 5813	A number is less than itself plus 1.
|- (A e. RR -> A < (A + 1))
Theorem	lep1t 5814	A number is less than or equal to itself plus 1.
|- (A e. RR -> A <_ (A + 1))
Theorem	ltp1 5815	A number is less than itself plus 1.
|- A e. RR   =>   |- A < (A + 1)
Theorem	recgt0i 5816	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21.
|- A e. RR   &   |- 0 < A   =>   |- 0 < (1 / A)
Theorem	ltm1t 5817	A number minus 1 is less than itself.
|- (A e. RR -> (A - 1) < A)
Theorem	letrp1t 5818	A transitive property of 'less than or equal' and plus 1.
|- ((A e. RR /\ B e. RR /\ A <_ B) -> A <_ (B + 1))
Theorem	p1let 5819	A transitive property of plus 1 and 'less than or equal'.
|- ((A e. RR /\ B e. RR /\ (A + 1) <_ B) -> A <_ B)
Theorem	prodgt0lem 5820	Lemma for prodgt0 5821.
Theorem	prodgt0 5821	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	prodge0 5822	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	ltmul1i 5823	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	ltmul1 5824	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	ltdiv1i 5825	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	ltdiv1 5826	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	ltmuldiv 5827	'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	prodgt0t 5828	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	prodgt02t 5829	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	prodge0t 5830	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	prodge02t 5831	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	ltmul1t 5832	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	ltmul2t 5833	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	lemul1t 5834	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	lemul2t 5835	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	ltmul2 5836	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 5837	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 5838	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 5839	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 5840	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 5841	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 5842	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 5843	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 5844	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 5845	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 5846	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 5847	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 5848	Multiplication by a number greater than 1.
|- ((A e. RR /\ B e. RR /\ 0 < A) -> (1 < B <-> A < (A x. B)))
Theorem	ltmulgt12t 5849	Multiplication by a number greater than 1.
|- ((A e. RR /\ B e. RR /\ 0 < A) -> (1 < B <-> A < (B x. A)))
Theorem	lemulge11t 5850	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 5851	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	ltdiv1tOLD 5852	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