mmtheorems65 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6401-6500 - Page 65 of 108

Type	Label	Description
Statement
Theorem	repos 6401	Two ways of saying that a real number is positive.
|- (A e. (0(,) +oo) <-> (A e. RR /\ 0 < A))
Theorem	ioof 6402	The set of open intervals of extended reals maps to subsets of reals.
|- (,):(RR* X. RR*)-->P~RR
Theorem	iccf 6403	The set of closed intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.)
|- [,]:(RR* X. RR*)-->P~RR*
Theorem	unirnioo 6404	The union of the range of the open interval function.
|- U.ran (,) = RR
Theorem	dfioo2 6405	Alternate definition of the set of open intervals of extended reals.
|- (,) = {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR | (x < w /\ w < y)})}
Theorem	lbicc2t 6406	The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.)
|- ((A e. RR /\ B e. RR /\ A <_ B) -> A e. (A[,]B))
Theorem	ubicc2t 6407	The upper bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.)
|- ((A e. RR /\ B e. RR /\ A <_ B) -> B e. (A[,]B))
Theorem	ioonegt 6408	Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (C e. (A(,)B) <-> -uC e. (-uB(,)-uA)))
Theorem	iccnegt 6409	Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (C e. (A[,]B) <-> -uC e. (-uB[,]-uA)))
Theorem	icoshft 6410	A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (X e. (A[,)B) -> (X + C) e. ((A + C)[,)(B + C))))
Theorem	icoshftf1oi 6411	Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.)
|- F = {<.x, y>. | (x e. (A[,)B) /\ y = (x + C))}   &   |- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- F:(A[,)B)-1-1-onto->((A + C)[,)(B + C))
Theorem	icoshftf1olem 6412	Lemma for icoshftf1o 6413.
Theorem	icoshftf1o 6413	Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.)
|- F = {<.x, y>. | (x e. (A[,)B) /\ y = (x + C))}   =>   |- ((A e. RR /\ B e. RR /\ C e. RR) -> F:(A[,)B)-1-1-onto->((A + C)[,)(B + C)))
Theorem	icounlem 6414	Lemma for icoun 6415.
Theorem	icoun 6415	The union of end-to-end closed-below, open-above real intervals. (Contributed by Paul Chapman, 15-Mar-2008.)
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A <_ B /\ B <_ C)) -> ((A[,)B) u. (B[,)C)) = (A[,)C))
Theorem	snunioolem 6416	Lemma for snunioo 6417.
Theorem	snunioo 6417	The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.)
|- ((A e. RR /\ B e. RR /\ A < B) -> ({A} u. (A(,)B)) = (A[,)B))
Theorem	ioojoint 6418	Join two open intervals to create a third.
|- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> (((A(,)B) u. {B}) u. (B(,)C)) = (A(,)C))
Upper partititions of integers
Syntax	cuz 6419	Extend class notation with the upper integer function. Read "ZZ>` M" as "the set of integers greater than or equal to M."
class ZZ>
Definition	df-uz 6420	Define a function whose value at j is the semi-infinite set of contiguous integers starting at j, which we will also call the upper integers starting at j. Read "ZZ>` M" as "the set of integers greater than or equal to M." See uzvalt 6421 for its value, uzssz 6432 for its relationship to ZZ, nnuz 6441 and nn0uz 6440 for its relationships to NN and NN0, and eluz1t 6422 and eluz2t 6423 for its membership relations.
|- ZZ> = {<.j, y>. | (j e. ZZ /\ y = {k e. ZZ | j <_ k})}
Theorem	uzvalt 6421	The value of the upper integers function.
|- (N e. ZZ -> (ZZ>` N) = {k e. ZZ | N <_ k})
Theorem	eluz1t 6422	Membership in the set of upper integers starting at M.
|- (M e. ZZ -> (N e. (ZZ>` M) <-> (N e. ZZ /\ M <_ N)))
Theorem	eluz2t 6423	Membership in a set of upper integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show M e. ZZ.
|- (N e. (ZZ>` M) <-> (M e. ZZ /\ N e. ZZ /\ M <_ N))
Theorem	eluz1 6424	Membership in a set of upper integers.
|- M e. ZZ   =>   |- (N e. (ZZ>` M) <-> (N e. ZZ /\ M <_ N))
Theorem	eluzelz 6425	Implication of membership in a set of upper integers.
|- (N e. (ZZ>` M) -> N e. ZZ)
Theorem	eluzel2 6426	Implication of membership in a set of upper integers.
|- (N e. (ZZ>` M) -> M e. ZZ)
Theorem	eluzle 6427	Implication of membership in a set of upper integers.
|- (N e. (ZZ>` M) -> M <_ N)
Theorem	eluzt 6428	Membership in a set of upper integers.
|- ((M e. ZZ /\ N e. ZZ) -> (N e. (ZZ>` M) <-> M <_ N))
Theorem	uzidt 6429	Membership of the least member in a set of upper integers.
|- (M e. ZZ -> M e. (ZZ>` M))
Theorem	uztrn 6430	Transitive law for sets of upper integers.
|- ((M e. (ZZ>` K) /\ K e. (ZZ>` N)) -> M e. (ZZ>` N))
Theorem	uznegit 6431	Contraposition law for upper integers.
|- (N e. (ZZ>` M) -> -uM e. (ZZ>` -uN))
Theorem	uzssz 6432	A set of upper integers is a subset of all integers.
|- (ZZ>` M) (_ ZZ
Theorem	uzss 6433	Subset relationship for two sets of upper integers.
|- (N e. (ZZ>` M) -> (ZZ>` N) (_ (ZZ>` M))
Theorem	uz11t 6434	The upper integers function is one-to-one.
|- (M e. ZZ -> ((ZZ>` M) = (ZZ>` N) <-> M = N))
Theorem	eluzp1m1t 6435	Membership in the next set of upper integers.
|- ((M e. ZZ /\ N e. (ZZ>` (M + 1))) -> (N - 1) e. (ZZ>` M))
Theorem	eluzp1lt 6436	Strict ordering implied by membership in the next set of upper integers.
|- ((M e. ZZ /\ N e. (ZZ>` (M + 1))) -> M < N)
Theorem	eluzp1p1t 6437	Membership in the next set of upper integers.
|- (N e. (ZZ>` M) -> (N + 1) e. (ZZ>` (M + 1)))
Theorem	eluzaddi 6438	Membership in a later set of upper integers. (Contributed by Paul Chapman, 22-Nov-2007.)
|- M e. ZZ   &   |- K e. ZZ   =>   |- (N e. (ZZ>` M) -> (N + K) e. (ZZ>` (M + K)))
Theorem	eluzsubi 6439	Membership in an earlier set of upper integers. (Contributed by Paul Chapman, 22-Nov-2007.)
|- M e. ZZ   &   |- K e. ZZ   =>   |- (N e. (ZZ>` (M + K)) -> (N - K) e. (ZZ>` M))
Theorem	nn0uz 6440	Nonnegative integers expressed as a set of upper integers.
|- NN0 = (ZZ>` 0)
Theorem	nnuz 6441	Natural numbers expressed as a set of upper integers.
|- NN = (ZZ>` 1)
Theorem	elnnuz 6442	A natural number expressed as a member of a set of upper integers.
|- (N e. NN <-> N e. (ZZ>` 1))
Theorem	elnn0uz 6443	A nonnegative integer expressed as a member a set of upper integers.
|- (N e. NN0 <-> N e. (ZZ>` 0))
Theorem	raluz 6444	Restricted universal quantification in a set of upper integers.
|- (M e. ZZ -> (A.n e. (ZZ>` M)ph <-> A.n e. ZZ (M <_ n -> ph)))
Theorem	raluz2 6445	Restricted universal quantification in a set of upper integers.
|- (A.n e. (ZZ>` M)ph <-> (M e. ZZ -> A.n e. ZZ (M <_ n -> ph)))
Theorem	rexuz 6446	Restricted existential quantification in a set of upper integers.
|- (M e. ZZ -> (E.n e. (ZZ>` M)ph <-> E.n e. ZZ (M <_ n /\ ph)))
Theorem	rexuz2 6447	Restricted existential quantification in a set of upper integers.
|- (E.n e. (ZZ>` M)ph <-> (M e. ZZ /\ E.n e. ZZ (M <_ n /\ ph)))
Theorem	2rexuz 6448	Double existential quantification in a set of upper integers.
|- (E.mE.n e. (ZZ>` m)ph <-> E.m e. ZZ E.n e. ZZ (m <_ n /\ ph))
Theorem	peano2uz 6449	Second Peano postulate for a set of upper integers.
|- (N e. (ZZ>` M) -> (N + 1) e. (ZZ>` M))
Theorem	peano2uzr 6450	Reversed second Peano axiom for upper integers.
|- ((M e. ZZ /\ N e. (ZZ>` (M + 1))) -> N e. (ZZ>` M))
Theorem	uzaddclt 6451	Addition closure law for a set of upper integers.
|- ((N e. (ZZ>` M) /\ K e. NN0) -> (N + K) e. (ZZ>` M))
Theorem	uzind4 6452	Induction on the set of upper integers that starts at an integer M. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction hypothesis.
|- (j = M -> (ph <-> ps))   &   |- (j = k -> (ph <-> ch))   &   |- (j = (k + 1) -> (ph <-> th))   &   |- (j = N -> (ph <-> ta))   &   |- (M e. ZZ -> ps)   &   |- (k e. (ZZ>` M) -> (ch -> th))   =>   |- (N e. (ZZ>` M) -> ta)
Theorem	uzind4ALT 6453	Alternate version of uzind4 6452 with different hypothesis order for easier use with the Metamath Proof Assistant, since "assign last" will assign the substitutions first. (This may or may not be kept permanenently, or it may replace uzind4 6452 - I haven't decided yet. -nm)
|- (M e. ZZ -> ps)   &   |- (k e. (ZZ>` M) -> (ch -> th))   &   |- (j = M -> (ph <-> ps))   &   |- (j = k -> (ph <-> ch))   &   |- (j = (k + 1) -> (ph <-> th))   &   |- (j = N -> (ph <-> ta))   =>   |- (N e. (ZZ>` M) -> ta)
Theorem	uzind4s 6454	Induction on the set of upper integers that starts at an integer M, using explicit substitution. The hypotheses are the basis and the induction hypothesis.
|- (M e. ZZ -> [M / k]ph)   &   |- (k e. (ZZ>` M) -> (ph -> [(k + 1) / k]ph))   =>   |- (N e. (ZZ>` M) -> [N / k]ph)
Theorem	uzind4s2 6455	Induction on the set of upper integers that starts at an integer M, using explicit substitution. The hypotheses are the basis and the induction hypothesis. Use this instead of uzind4s 6454 when j and k must be distinct in [(k + 1) / j]ph.
|- (M e. ZZ -> [M / j]ph)   &   |- (k e. (ZZ>` M) -> ([k / j]ph -> [(k + 1