mmtheorems60 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5901-6000 - Page 60 of 108

Type	Label	Description
Statement
Theorem	ltdiv2t 5901	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	ltrec1t 5902	Reciprocal swap in a 'less than' relation.
|- (((A e. RR /\ 0 < A) /\ (B e. RR /\ 0 < B)) -> ((1 / A) < B <-> (1 / B) < A))
Theorem	lerec2t 5903	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	ledivdivt 5904	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	lediv2t 5905	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	ltdiv23t 5906	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	lediv23t 5907	Swap denominator with other side of 'less than or equal to'.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ (0 < B /\ 0 < C)) -> ((A / B) <_ C <-> (A / C) <_ B))
Theorem	ltdiv23 5908	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	ltdiv23i 5909	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	lediv12it 5910	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	lediv2it 5911	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	reclt1t 5912	The reciprocal of a positive number less than 1 is greater than 1.
|- ((A e. RR /\ 0 < A) -> (A < 1 <-> 1 < (1 / A)))
Theorem	recgt1t 5913	The reciprocal of a positive number greater than 1 is less than 1.
|- ((A e. RR /\ 0 < A) -> (1 < A <-> (1 / A) < 1))
Theorem	recgt1it 5914	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 5915	Construct a number less than 1 from any nonnegative number.
|- ((A e. RR /\ 0 <_ A) -> (A / (1 + A)) < 1)
Theorem	recrecltt 5916	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	le2msqt 5917	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	halfpos 5918	A positive number is greater than its half.
|- A e. RR   =>   |- (0 < A <-> (A / (1 + 1)) < A)
Theorem	ledivp1t 5919	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	ledivp1 5920	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	ltdivp1 5921	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	posex 5922	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 5923	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 5924	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 5925	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 5926	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	xrmaxltt 5927	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	xrltmint 5928	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 5929	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 5930	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 5931	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	maxlet 5932	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 5933	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 5934	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	lemint 5935	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	maxltt 5936	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	ltmint 5937	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 5938	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 5939	The natural numbers of analysis start at one (unlike the ordinal natural numbers, i.e. the members of the set om, df-om 3141, 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 5951 for the principle of mathematical induction. See dfnn2 5950 for a slight variant. See df-n0 6114 for the set of nonnegative integers NN0 starting at zero. See dfn2 6126 for NN defined in terms of NN0.
|- NN = |^|{x | (1 e. x /\ A.y e. x (y + 1) e. x)}
Theorem	peano5nn 5940	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 5941	The natural numbers are a subset of the reals.
|- NN (_ RR
Theorem	nnsscn 5942	The natural numbers are a subset of the complex numbers.
|- NN (_ CC
Theorem	nnret 5943	A natural number is a real number.
|- (A e. NN -> A e. RR)
Theorem	nncnt 5944	A natural number is a complex number.
|- (A e. NN -> A e. CC)
Theorem	nnre 5945	A natural number is a real number.
|- A e. NN   =>   |- A e. RR
Theorem	nncn 5946	A natural number is a complex number.
|- A e. NN   =>   |- A e. CC
Theorem	nnex 5947	The set of natural numbers exists.
|- NN e. V
Theorem	1nn 5948	Peano postulate: 1 is a natural number.
|- 1 e. NN
Theorem	peano2nn 5949	Peano postulate: a successor of a natural number is a natural number.
|- (A e. NN -> (A + 1) e. NN)
Theorem	dfnn2 5950	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 5951	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 nnaddclt 5954 for