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	recp1lt1 5901	Construct a number less than 1 from any nonnegative number.
|- ((A e. RR /\ 0 <_ A) -> (A / (1 + A)) < 1)
Theorem	recrecltt 5902	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 5903	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 5904	A positive number is greater than its half.
|- A e. RR   =>   |- (0 < A <-> (A / (1 + 1)) < A)
Theorem	ledivp1t 5905	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 5906	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 5907	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 5908	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 5909	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 5910	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 5911	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 5912	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 5913	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 5914	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 5915	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 5916	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 5917	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 5918	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 5919	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 5920	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 5921	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 5922	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 5923	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 5924	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 5925	The natural numbers of analysis start at one (unlike the ordinal natural numbers, i.e. the members of the set om, df-om 3132, 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 5937 for the principle of mathematical induction. See dfnn2 5936 for a slight variant. See df-n0 6100 for the set of nonnegative integers NN0 starting at zero. See dfn2 6112 for NN defined in terms of NN0.
|- NN = |^|{x | (1 e. x /\ A.y e. x (y + 1) e. x)}
Theorem	peano5nn 5926	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 5927	The natural numbers are a subset of the reals.
|- NN (_ RR
Theorem	nnsscn 5928	The natural numbers are a subset of the complex numbers.
|- NN (_ CC
Theorem	nnret 5929	A natural number is a real number.
|- (A e. NN -> A e. RR)
Theorem	nncnt 5930	A natural number is a complex number.
|- (A e. NN -> A e. CC)
Theorem	nnre 5931	A natural number is a real number.
|- A e. NN   =>   |- A e. RR
Theorem	nncn 5932	A natural number is a complex number.
|- A e. NN   =>   |- A e. CC
Theorem	nnex 5933	The set of natural numbers exists.
|- NN e. V
Theorem	1nn 5934	Peano postulate: 1 is a natural number.
|- 1 e. NN
Theorem	peano2nn 5935	Peano postulate: a successor of a natural number is a natural number.
|- (A e. NN -> (A + 1) e. NN)
Theorem	dfnn2 5936	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 5937	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 5940 for an example of its use. See nn0ind 6212 for induction on nonnegative integers and uzind 6205, uzind4 6450 for induction on an arbitrary set of upper integers. See indstr 6461 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 5938	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 5937 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 5939	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	nnaddclt 5940	Closure of addition of natural numbers, proved by induction on the second addend.
|- ((A e. NN /\ B e. NN) -> (A + B) e. NN)
Theorem	nnmulclt 5941	Closure of multiplication of natural numbers.
|- ((A e. NN /\ B e. NN) -> (A x. B) e. NN)
Theorem	nn2get 5942	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	nnge1t 5943	A natural number is one or greater.
|- (A e. NN -> 1 <_ A)
Theorem	nngt1ne1t 5944	A natural number is greater than one iff it is not equal to one.
|- (A e. NN -> (1 < A <-> A =/= 1))
Theorem	nnle1eq1t 5945	A natural number is less than or equal to one iff it is equal to one.
|- (A e. NN -> (A <_ 1 <-> A = 1))
Theorem	nngt0t 5946	A natural number is positive.
|- (A e. NN -> 0 < A)
Theorem	lt1nnn 5947	A number less than one is not a natural number.
|- ((A e. RR /\ A < 1) -> -. A e. NN)
Theorem	0nnn 5948	Zero is not a natural number.
|- -. 0 e. NN
Theorem	nnne0t 5949	A natural number is non-zero.
|- (A e. NN -> A =/= 0)
Theorem	nngt0 5950	A natural number is positive (inference version).
|- A e. NN   =>   |- 0 < A
Theorem	nnne0 5951	A natural number is non-zero (inference version).
|- A e. NN   =>   |- A =/= 0
Theorem	nnrecret 5952	The reciprocal of a natural number is real.
|- (N e. NN -> (1 / N) e. RR)
Theorem	nnrecgt0t 5953	The reciprocal of a natural number is positive.
|- (A e. NN -> 0 < (1 / A))
Theorem	nnleltp1t 5954	Natural number ordering relation.
|- ((A e. NN /\ B e. NN) -> (A <_ B <-> A < (B + 1)))
Theorem	nnltp1let 5955	Natural number ordering relation.
|- ((A e. NN /\ B e. NN) -> (A < B <-> (A + 1) <_ B))
Theorem	nnsub 5956	Subtraction of natural numbers.
|- A e. NN   &   |- B e. NN   =>   |- (A < B <-> (B - A) e. NN)
Theorem	nnsubt 5957	Subtraction of natural numbers.
|- ((A e. NN /\ B e. NN) -> (A < B <-> (B - A) e. NN))
Theorem	nnaddm1clt 5958	Closure of addition of natural numbers minus one.
|- ((A e. NN /\ B e. NN) -> ((A + B) - 1) e. NN)
Theorem	nndivt 5959	Two ways to express "A divides B" for natural numbers.
|- ((A