mmtheorems61 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6001-6100 - Page 61 of 123

Type	Label	Description
Statement
Theorem	divgt0i 6001	The ratio of two positive numbers is positive.
|- A e. RR   &   |- B e. RR   =>   |- ((0 < A /\ 0 < B) -> 0 < (A / B))
Theorem	divge0i 6002	The ratio of nonnegative and positive numbers is nonnegative.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ 0 < B) -> 0 <_ (A / B))
Theorem	divgt0i2i 6003	The ratio of two positive numbers is positive.
|- A e. RR   &   |- B e. RR   &   |- 0 < B   =>   |- (0 < A -> 0 < (A / B))
Theorem	divgt0ii 6004	The ratio of two positive numbers is positive.
|- A e. RR   &   |- B e. RR   &   |- 0 < A   &   |- 0 < B   =>   |- 0 < (A / B)
Theorem	recgt0 6005	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21.
|- ((A e. RR /\ 0 < A) -> 0 < (1 / A))
Theorem	recgt0i 6006	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21.
|- A e. RR   =>   |- (0 < A -> 0 < (1 / A))
Theorem	ltmuldiv 6007	'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	ltmuldivOLD 6008	'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	ltmuldiv2 6009	'Less than' relationship between division and multiplication.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> ((C x. A) < B <-> A < (B / C)))
Theorem	ltmuldiv2OLD 6010	'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	ltdivmul 6011	'Less than' relationship between division and multiplication.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> ((A / C) < B <-> A < (C x. B)))
Theorem	ltdivmulOLD 6012	'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	ledivmul 6013	'Less than or equal to' relationship between division and multiplication.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> ((A / C) <_ B <-> A <_ (C x. B)))
Theorem	ledivmulOLD 6014	'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	ltdivmul2 6015	'Less than' relationship between division and multiplication.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> ((A / C) < B <-> A < (B x. C)))
Theorem	lt2mul2div 6016	'Less than' relationship between division and multiplication.
|- (((A e. RR /\ (B e. RR /\ 0 < B)) /\ (C e. RR /\ (D e. RR /\ 0 < D))) -> ((A x. B) < (C x. D) <-> (A / D) < (C / B)))
Theorem	lt2mul2divOLD 6017	'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	ledivmul2 6018	'Less than or equal to' relationship between division and multiplication.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> ((A / C) <_ B <-> A <_ (B x. C)))
Theorem	ledivmul2OLD 6019	'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	lemuldiv 6020	'Less than or equal' relationship between division and multiplication.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> ((A x. C) <_ B <-> A <_ (B / C)))
Theorem	lemuldiv2 6021	'Less than or equal' relationship between division and multiplication.
|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> ((C x. A) <_ B <-> A <_ (B / C)))
Theorem	lemuldiv2OLD 6022	'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	ltrecii 6023	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	ltreci 6024	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	lereci 6025	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	lt2msqi 6026	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	le2msqi 6027	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	msq11i 6028	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	ltrec 6029	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	lerec 6030	The reciprocal of both sides of 'less than or equal to'.
|- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) -> (A <_ B <-> (1 / B) <_ (1 / A)))
Theorem	lt2msq 6031	Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.)
|- (((A e. RR /\ 0 <_ A) /\ (B e. RR /\ 0 <_ B)) -> (A < B <-> (A x. A) < (B x. B)))
Theorem	ltdiv2 6032	Division of a positive number by both sides of 'less than'.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ (0 < A /\ 0 < B /\ 0 < C)) -> (A < B <-> (C / B) < (C / A)))
Theorem	ltrec1 6033	Reciprocal swap in a 'less than' relation.
|- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) -> ((1 / A) < B <-> (1 / B) < A))
Theorem	lerec2 6034	Reciprocal swap in a 'less than or equal to' relation.
|- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) -> (A <_ (1 / B) <-> B <_ (1 / A)))
Theorem	ledivdiv 6035	Invert ratios of positive numbers and swap their ordering.
|- ((((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) /\ ((C e. RR /\ 0 < C) /\ (D e. RR /\ 0 < D))) -> ((A / B) <_ (C / D) <-> (D / C) <_ (B / A)))
Theorem	lediv2 6036	Division of a positive number by both sides of 'less than or equal to'.
|- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> (A <_ B <-> (C / B) <_ (C / A)))
Theorem	ltdiv23 6037	Swap denominator with other side of 'less than'.
|- ((A e. RR /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> ((A / B) < C <-> (A / C) < B))
Theorem	lediv23 6038	Swap denominator with other side of 'less than or equal to'.
|- ((A e. RR /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 < C)) -> ((A / B) <_ C <-> (A / C) <_ B))
Theorem	ltdiv23i 6039	Swap denominator with other side of 'less than'.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- ((0 < B /\ 0 < C) -> ((A / B) < C <-> (A / C) < B))
Theorem	ltdiv23ii 6040	Swap denominator with other side of 'less than'.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   &   |- 0 < B   &   |- 0 < C   =>   |- ((A / B) < C <-> (A / C) < B)
Theorem	lediv12a 6041	Comparison of ratio 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 / D) <_ (B / C))
Theorem	lediv2a 6042	Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
|- ((((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B) /\ (C e. RR /\ 0 <_ C)) /\ A <_ B) -> (C / B) <_ (C / A))
Theorem	reclt1 6043	The reciprocal of a positive number less than 1 is greater than 1.
|- ((A e. RR /\ 0 < A) -> (A < 1 <-> 1 < (1 / A)))
Theorem	recgt1 6044	The reciprocal of a positive number greater than 1 is less than 1.
|- ((A e. RR /\ 0 < A) -> (1 < A <-> (1 / A) < 1))
Theorem	recgt1i 6045	The reciprocal of a number greater than 1 is positive and less than 1.
|- ((A e. RR /\ 1 < A) -> (0 < (1 / A) /\ (1 / A) < 1))
Theorem	recp1lt1 6046	Construct a number less than 1 from any nonnegative number.
|- ((A e. RR /\ 0 <_ A) -> (A / (1 + A)) < 1)
Theorem	recreclt 6047	Given a positive number A, construct a new positive number less than both A and 1.
|- ((A e. RR /\ 0 < A) -> ((1 / (1 + (1 / A))) < 1 /\ (1 / (1 + (1 / A))) < A))
Theorem	le2msq 6048	The square function on nonnegative reals is monotonic.
|- (((A e. RR /\ 0 <_ A) /\ (B e. RR /\ 0 <_ B)) -> (A <_ B <-> (A x. A) <_ (B x. B)))
Theorem	halfposi 6049	A positive number is greater than its half.
|- A e. RR   =>   |- (0 < A <-> (A / (1 + 1)) < A)
Theorem	ledivp1 6050	Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.)
|- (((A e. RR /\ 0 <_ A) /\ (B e. RR /\ 0 <_ B)) -> ((A / (B + 1)) x. B) <_ A)
Theorem	ledivp1i 6051	Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.)
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- ((0 <_ A /\ 0 <_ C /\ A <_ (B / (C + 1))) -> (A x. C) <_ B)
Theorem	ltdivp1i 6052	Less-than and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.)
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- ((0 <_ A /\ 0 <_ C /\ A < (B / (C + 1))) -> (A x. C) < B)
Theorem	posexi 6053	There exists a positive number less than two others.
|- A e. RR   &   |- B e. RR   &   |- 0 < A   &   |- 0 < B   =>   |- E.x e. RR (0 < x /\ (x < A /\ x < B))
Theorem	xrmax1 6054	An extended real is less than or equal to the maximum of it and another.
|- ((A e. RR* /\ B e. RR*) -> A <_ if(A <_ B, B, A))
Theorem	xrmax2 6055	An extended real is less than or equal to the maximum of it and another.
|- ((A e. RR* /\ B e. RR*) -> B <_ if(A <_ B, B, A))
Theorem	xrmin1 6056	The minimum of two extended reals is less than or equal to one of them.
|- ((A e. RR* /\ B e. RR*) -> if(A <_ B, A, B) <_ A)
Theorem	xrmin2 6057	The minimum of two extended reals is less than or equal to one of them.
|- ((A e. RR* /\ B e. RR*) -> if(A <_ B, A, B) <_ B)
Theorem	xrmaxlt 6058	Two ways of saying the maximum of two extended reals is less than a third.
|- ((A e. RR* /\ B e. RR* /\ C e. RR*) -> (if(A <_ B, B, A) < C <-> (A < C /\ B < C)))
Theorem	xrltmin 6059	Two ways of saying an extended real is less than the minimum of two others.
|- ((A e. RR* /\ B e. RR* /\ C e. RR*) -> (A < if(B <_ C, B, C) <-> (A < B /\ A < C)))
Theorem	max1 6060	A number is less than or equal to the maximum of it and another.
|- ((A e. RR /\ B e. RR) -> A <_ if(A <_ B, B, A))
Theorem	max1ALT 6061	A number is less than or equal to the maximum of it and another.
|- (A e. RR -> A <_ if(A <_ B, B, A))
Theorem	max2 6062	A number is less than or equal to the maximum of it and another.
|- ((A e. RR /\ B e. RR) -> B <_ if(A <_ B, B, A))
Theorem	maxle 6063	Two ways of saying the maximum of two numbers is less than or equal to a third.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (if(A <_ B, B, A) <_ C <-> (A <_ C /\ B <_ C)))
Theorem	min1 6064	The minimum of two numbers is less than or equal to the first.
|- ((A e. RR /\ B e. RR) -> if(A <_ B, A, B) <_ A)
Theorem	min2 6065	The minimum of two numbers is less than or equal to the second.
|- ((A e. RR /\ B e. RR) -> if(A <_ B, A, B) <_ B)
Theorem	lemin 6066	Two ways of saying a number is less than or equal to the minimum of two others.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A <_ if(B <_ C, B, C) <-> (A <_ B /\ A <_ C)))
Theorem	maxlt 6067	Two ways of saying the maximum of two numbers is less than a third.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (if(A <_ B, B, A) < C <-> (A < C /\ B < C)))
Theorem	ltmin 6068	Two ways of saying a number is less than the minimum of two others.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A < if(B <_ C, B, C) <-> (A < B /\ A < C)))
Theorem	squeeze0 6069	If a nonnegative number is less than any positive number, it is zero.
|- ((A e. RR /\ 0 <_ A /\ A.x e. RR (0 < x -> A < x)) -> A = 0)
Natural numbers (as a subset of complex numbers)
Definition	df-n 6070	The natural numbers of analysis start at one (unlike the ordinal natural numbers, i.e. the members of the set om, df-om 3219, which start at zero). This is the convention used by most analysis books, and it is often convenient in proofs because we don't have to worry about division by zero. See nnind 6082 for the principle of mathematical induction. See dfnn2 6081 for a slight variant. See df-n0 6268 for the set of nonnegative integers NN0 starting at zero. See dfn2 6280 for NN defined in terms of NN0.
|- NN = |^|{x | (1 e. x /\ A.y e. x (y + 1) e. x)}
Theorem	peano5nni 6071	Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34.
|- A e. V   =>   |- ((1 e. A /\ A.x e. A (x + 1) e. A) -> NN (_ A)
Theorem	nnssre 6072	The natural numbers are a subset of the reals.
|- NN (_ RR
Theorem	nnsscn 6073	The natural numbers are a subset of the complex numbers.
|- NN (_ CC
Theorem	nnre 6074	A natural number is a real number.
|- (A e. NN -> A e. RR)
Theorem	nncn 6075	A natural number is a complex number.
|- (A e. NN -> A e. CC)
Theorem	nnrei 6076	A natural number is a real number.
|- A e. NN   =>   |- A e. RR
Theorem	nncni 6077	A natural number is a complex number.
|- A e. NN   =>   |- A e. CC
Theorem	nnex 6078	The set of natural numbers exists.
|- NN e. V
Theorem	1nn 6079	Peano postulate: 1 is a natural number.
|- 1 e. NN
Theorem	peano2nn 6080	Peano postulate: a successor of a natural number is a natural number.
|- (A e. NN -> (A + 1) e. NN)
Theorem	dfnn2 6081	Alternate definition of the set of natural numbers. Definition of positive integers in [Apostol] p. 22.
|- NN = |^|{x | (x (_ RR /\ 1 e. x /\ A.y e. x (y + 1) e. x)}
Principle of mathematical induction
Theorem	nnind 6082	Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. See nnaddcl 6085 for an example of its use. See nn0ind 6383 for induction on nonnegative integers and uzind 6376, uzind4 6577 for induction on an arbitrary set of upper integers. See indstr 6588 for strong induction.
|- (x = 1 -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = (y + 1) -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   &   |- ps   &   |- (y e. NN -> (ch -> th))   =>   |- (A e. NN -> ta)
Theorem	nnindALT 6083	Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction hypothesis and the basis. (This ALT version of nnind 6082 is easier to use with the Proof Assistant since 'assign last' will be applied to the substitution instances first. We may switch to it as the official version.)
|- (y e. NN -> (ch -> th))   &   |- ps   &   |- (x = 1 -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = (y + 1) -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   =>   |- (A e. NN -> ta)
Natural numbers (cont.)
Theorem	nn1suc 6084	If a statement holds for 1 and also holds for a successor, it holds for all natural numbers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor.
|- (x = 1 -> (ph <-> ps))   &   |- (x = (y + 1) -> (ph <-> ch))   &   |- (x = A -> (ph <-> th))   &   |- ps   &   |- (y e. NN -> ch)   =>   |- (A e. NN -> th)
Theorem	nnaddcl 6085	Closure of addition of natural numbers, proved by induction on the second addend.
|- ((A e. NN /\ B e. NN) -> (A + B) e. NN)
Theorem	nnmulcl 6086	Closure of multiplication of natural numbers.
|- ((A e. NN /\ B e. NN) -> (A x. B) e. NN)
Theorem	nn2ge 6087	There exists a natural number greater than or equal to any two others.
|- ((A e. NN /\ B e. NN) -> E.x e. NN (A <_ x /\ B <_ x))
Theorem	nnge1 6088	A natural number is one or greater.
|- (A e. NN -> 1 <_ A)
Theorem	nngt1ne1 6089	A natural number is greater than one iff it is not equal to one.
|- (A e. NN -> (1 < A <-> A =/= 1))
Theorem	nnle1eq1 6090	A natural number is less than or equal to one iff it is equal to one.
|- (A e. NN -> (A <_ 1 <-> A = 1))
Theorem	nngt0 6091	A natural number is positive.
|- (A e. NN -> 0 < A)
Theorem	lt1nnn 6092	A number less than one is not a natural number.
|- ((A e. RR /\ A < 1) -> -. A e. NN)
Theorem	0nnn 6093	Zero is not a natural number.
|- -. 0 e. NN
Theorem	nnne0 6094	A natural number is non-zero.
|- (A e. NN -> A =/= 0)
Theorem	nngt0i 6095	A natural number is positive (inference version).
|- A e. NN   =>   |- 0 < A
Theorem	nnne0i 6096	A natural number is non-zero (inference version).
|- A e. NN   =>   |- A =/= 0
Theorem	nndivre 6097	The quotient of a real and a natural number is real.
|- ((A e. RR /\ N e. NN) -> (A / N) e. RR)
Theorem	nnrecre 6098	The reciprocal of a natural number is real.
|- (N e. NN -> (1 / N) e. RR)
Theorem	nnrecgt0 6099	The reciprocal of a natural number is positive.
|- (A e. NN -> 0 < (1 / A))
Theorem	nnleltp1 6100	Natural number ordering relation.
|- ((A e. NN /\ B e. NN) -> (A <_ B <-> A < (B + 1)))