mmtheorems66 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6501-6600 - Page 66 of 108

Type	Label	Description
Statement
Theorem	elfzuz2t 6501	Implication of membership in a finite set of sequential integers.
|- ((N e. A /\ K e. (M...N)) -> N e. (ZZ>` M))
Theorem	eluzfz1t 6502	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>` M) -> M e. (M...N))
Theorem	elfzuzt 6503	A member of a finite set of sequential integers belongs to a set of upper integers.
|- (K e. (M...N) -> K e. (ZZ>` M))
Theorem	eluzfz2t 6504	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>` M) -> N e. (M...N))
Theorem	eluzfz2b 6505	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>` M) <-> N e. (M...N))
Theorem	elfz3t 6506	Membership in a finite set of sequential integers containing one integer.
|- (N e. ZZ -> N e. (N...N))
Theorem	elfz1eqt 6507	Membership in a finite set of sequential integers containing one integer.
|- (K e. (N...N) -> K = N)
Theorem	fznt 6508	A finite set of sequential integers is empty if the bounds are reversed.
|- ((M e. ZZ /\ N e. ZZ) -> (N < M <-> (M...N) = (/)))
Theorem	elfznnt 6509	A member of a finite set of sequential integers starting at 1 is a natural number.
|- (K e. (1...N) -> K e. NN)
Theorem	elfz2nn0t 6510	Membership in a finite set of sequential integers starting at 0.
|- (N e. A -> (K e. (0...N) <-> (K e. NN0 /\ N e. NN0 /\ K <_ N)))
Theorem	elfznn0t 6511	A member of a finite set of sequential integers starting at 0 is a nonnegative integer.
|- (K e. (0...N) -> K e. NN0)
Theorem	elfz3nn0t 6512	The upper bound of a nonempty finite set of sequential integers starting at 0 is a nonnegative integer.
|- ((N e. A /\ K e. (0...N)) -> N e. NN0)
Theorem	fznn0subt 6513	Subtraction closure for a member of a finite set of sequential integers.
|- ((N e. A /\ K e. (M...N)) -> (N - K) e. NN0)
Theorem	fznn0sub2t 6514	Subtraction closure for a member of a finite set of sequential integers.
|- ((N e. A /\ K e. (0...N)) -> (N - K) e. (0...N))
Theorem	fzaddelt 6515	Membership of a sum in a finite set of sequential integers.
|- (((M e. ZZ /\ N e. ZZ) /\ (J e. ZZ /\ K e. ZZ)) -> (J e. (M...N) <-> (J + K) e. ((M + K)...(N + K))))
Theorem	fzsubelt 6516	Membership of a difference in a finite set of sequential integers.
|- (((M e. ZZ /\ N e. ZZ) /\ (J e. ZZ /\ K e. ZZ)) -> (J e. (M...N) <-> (J - K) e. ((M - K)...(N - K))))
Theorem	fzoptht 6517	A finite set of sequential integers can represent an ordered pair.
|- ((N e. (ZZ>` M) /\ K e. A) -> ((M...N) = (J...K) <-> (M = J /\ N = K)))
Theorem	fzss1t 6518	Subset relationship for finite sets of sequential integers.
|- ((K e. (ZZ>` M) /\ N e. ZZ) -> (K...N) (_ (M...N))
Theorem	fzss2t 6519	Subset relationship for finite sets of sequential integers.
|- ((N e. (ZZ>` K) /\ M e. ZZ) -> (M...K) (_ (M...N))
Theorem	fzssuzt 6520	A finite set of sequential integers is a subset of a set of upper integers.
|- (M...N) (_ (ZZ>` M)
Theorem	fzssp1t 6521	Subset relationship for finite sets of sequential integers.
|- ((M e. ZZ /\ N e. ZZ) -> (M...N) (_ (M...(N + 1)))
Theorem	fzp1sst 6522	Subset relationship for finite sets of sequential integers.
|- ((M e. ZZ /\ N e. ZZ) -> ((M + 1)...N) (_ (M...N))
Theorem	fzelp1t 6523	Membership in a set of sequential integers with an appended element.
|- ((N e. A /\ K e. (M...N)) -> K e. (M...(N + 1)))
Theorem	fzelp1 6524	Membership in a set of sequential integers with an appended element.
|- N e. V   =>   |- (K e. (M...N) -> K e. (M...(N + 1)))
Theorem	elfzp1 6525	Append an element to a finite set of sequential integers.
|- (N e. (ZZ>` M) -> (K e. (M...(N + 1)) <-> (K e. (M...N) \/ K = (N + 1))))
Theorem	fzrevt 6526	Reversal of start and end of a finite set of sequential integers.
|- (((M e. ZZ /\ N e. ZZ) /\ (J e. ZZ /\ K e. ZZ)) -> (K e. ((J - N)...(J - M)) <-> (J - K) e. (M...N)))
Theorem	fzrev2t 6527	Reversal of start and end of a finite set of sequential integers.
|- (((M e. ZZ /\ N e. ZZ) /\ (J e. ZZ /\ K e. ZZ)) -> (K e. (M...N) <-> (J - K) e. ((J - N)...(J - M))))
Theorem	fzrev2it 6528	Reversal of start and end of a finite set of sequential integers.
|- ((N e. A /\ J e. ZZ /\ K e. (M...N)) -> (J - K) e. ((J - N)...(J - M)))
Theorem	fzrev3t 6529	The "complement" of a member of a finite set of sequential integers.
|- ((N e. A /\ K e. ZZ) -> (K e. (M...N) <-> ((M + N) - K) e. (M...N)))
Theorem	fzrev3it 6530	The "complement" of a member of a finite set of sequential integers.
|- ((N e. A /\ K e. (M...N)) -> ((M + N) - K) e. (M...N))
Theorem	fznn0t 6531	Finite set of sequential integers starting at 0.
|- (N e. NN0 -> (K e. (0...N) <-> (K e. NN0 /\ K <_ N)))
Theorem	fz1sbct 6532	Quantification over a one-member finite set of sequential integers in terms of substitution.
|- (N e. ZZ -> (A.k e. (N...N)ph <-> [N / k]ph))
Theorem	fzneuzt 6533	No finite set of sequential integers equals a set of upper integers.
|- ((N e. (ZZ>` M) /\ K e. ZZ) -> -. (M...N) = (ZZ>` K))
Theorem	fzrevralt 6534	Reversal of scanning order inside of a quantification over a finite set of sequential integers.
|- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (A.j e. (M...N)ph <-> A.k e. ((K - N)...(K - M))[(K - k) / j]ph))
Theorem	fzrevral2t 6535	Reversal of scanning order inside of a quantification over a finite set of sequential integers.
|- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (A.j e. ((K - N)...(K - M))ph <-> A.k e. (M...N)[(K - k) / j]ph))
Theorem	fzrevral3t 6536	Reversal of scanning order inside of a quantification over a finite set of sequential integers.
|- ((M e. ZZ /\ N e. ZZ) -> (A.j e. (M...N)ph <-> A.k e. (M...N)[((M + N) - k) / j]ph))
Theorem	fzshftralt 6537	Shift the scanning order inside of a quantification over a finite set of sequential integers.
|- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (A.j e. (M...N)ph <-> A.k e. ((M + K)...(N + K))[(k - K) / j]ph))
Theorem	fsequb 6538	The values of a finite real sequence have an upper bound. Warning: The HTML proof page is 1/2 megabyte in size.
|- ((N e. (ZZ>` M) /\ A.k e. (M...N)(F` k) e. RR) -> E.x e. RR A.k e. (M...N)(F` k) < x)
Theorem	fsequb2 6539	The values of a finite real sequence have an upper bound.
|- ((N e. (ZZ>` M) /\ F:(M...N)-->RR) -> E.x e. RR A.y e. ran F y <_ x)
Theorem	fseqsupcl 6540	The values of a finite real sequence have a supremum.
|- ((N e. (ZZ>` M) /\ F:(M...N)-->RR) -> sup(ran F, RR, < ) e. RR)
Theorem	fseqsupub 6541	The values of a finite real sequence are bounded by their supremum.
|- N e. V   =>   |- ((K e. (M...N) /\ F:(M...N)-->RR) -> (F` K) <_ sup(ran F, RR, < ))
Superior limit (lim sup)
Syntax	clsp 6542	Extend class notation to include the limsup function.
class limsup
Definition	df-limsup 6543	Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupvalt 6544 for its value.
|- limsup = {<.x, y>. | y = sup({z | E.k e. ZZ z = sup(((x"(ZZ>` k)) i^i RR*), RR*, < )}, RR*, `' < )}
Theorem	limsupvalt 6544	The superior limit of an infinite sequence F of extended real numbers, which is the infimum (indicated by `' <) of the set of suprema of all upper infinite subsequences of F. Definition 12-4.1 of [Gleason] p. 175.
|- (F e. A -> (limsup` F) = sup({x | E.k e. ZZ x = sup(((F"(ZZ>` k)) i^i RR*), RR*, < )}, RR*, `' < ))
Theorem	limsupclt 6545	Closure of the superior limit.
|- (F e. A -> (limsup` F) e. RR*)
Infinite sequence builders "seq" and "seq0"
Syntax	cseqz 6546	Extend class notation with arbitrarily-based recursive sequence builder.
class seq
Syntax	cseq0 6547	Extend class notation with 0-based recursive sequence builder.
class seq0
Definition	df-seqz 6548	Define a recursive sequence builder operation that starts at an arbitrary integer index. See seqz1 6562 and seqzp1 6563 for its initial and successor values. Theorems seq0seqz 6557 and seq1seqz 6556 derive the 0-based seq0 and the 1-based seq1 as special cases.
|- seq = {<.<.x, g>., h>. | h = ((((2nd` x) seq1 (g shift (1 - (1st` x)))) shift ((1st` x) - 1)) |` {k e. ZZ | (1st` x) <_ k})}
Definition	df-seq0 6549	Define a recursive sequence builder operation that starts at index 0. This is a frequently-used variation of the seq1 operation (see df-seq1 6323), which starts at index 1. See seq00 6565 and seq0p1 6566 for its initial and successor values. See dfseq0 6578 for an alternate definition.
|- seq0 = {<.<.f, g>., h>. | h = (((f seq1 (g shift 1)) shift -u1) |` NN0)}
Theorem	seq0fval 6550	Value of the 0-based recursive sequence builder operation.
|- S e. V   &   |- F e. V   =>   |- (S seq0 F) = (((S seq1 (F shift 1)) shift -u1) |` NN0)
Theorem	seq0valt 6551	Value of the 0-based recursive sequence builder operation.
|- S e. V   &   |- F e. V