mmtheorems65 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6401-6500 - Page 65 of 107

Type	Label	Description
Statement
Theorem	uzind4ALT 6401	Alternate version of uzind4 6400 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 6400 - 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 6402	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 6403	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 6402 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) / j]ph))   =>   |- (N e. (ZZ>` M) -> [N / j]ph)
Theorem	uzind4i 6404	Induction on the upper integers that start at M. The first hypothesis specifies the lower bound, the next four give us the substitution instances we need, and the last two are the basis and the induction hypothesis.
|- M e. ZZ   &   |- (j = M -> (ph <-> ps))   &   |- (j = k -> (ph <-> ch))   &   |- (j = (k + 1) -> (ph <-> th))   &   |- (j = N -> (ph <-> ta))   &   |- ps   &   |- (k e. (ZZ>` M) -> (ch -> th))   =>   |- (N e. (ZZ>` M) -> ta)
Theorem	uzwo 6405	Well-ordering principle: any non-empty subset of a set of upper integers has a least element.
|- ((S (_ (ZZ>` M) /\ S =/= (/)) -> E.j e. S A.k e. S j <_ k)
Theorem	uzwoOLD 6406	Well-ordering principle: any non-empty subset of the upper integers has a least element.
|- ((S (_ (ZZ>` M) /\ -. S = (/)) -> E.j e. S A.k e. S j <_ k)
Theorem	uzwo2 6407	Well-ordering principle: any non-empty subset of upper integers has a unique least element.
|- ((S (_ (ZZ>` M) /\ S =/= (/)) -> E!j e. S A.k e. S j <_ k)
Theorem	nnwo 6408	Well-ordering principle: any non-empty set of natural numbers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34.
|- ((A (_ NN /\ A =/= (/)) -> E.x e. A A.y e. A x <_ y)
Theorem	nnwof 6409	Well-ordering principle: any non-empty set of natural numbers has a least element. This version allows x and y to be present in A as long as they are effectively not free.
|- (z e. A -> A.x z e. A)   &   |- (z e. A -> A.y z e. A)   =>   |- ((A (_ NN /\ A =/= (/)) -> E.x e. A A.y e. A x <_ y)
Theorem	nnwos 6410	Well-ordering principle: any non-empty set of natural numbers has a least element (schema form).
|- (x = y -> (ph <-> ps))   =>   |- (E.x e. NN ph -> E.x e. NN (ph /\ A.y e. NN (ps -> x <_ y)))
Theorem	indstr 6411	Strong Mathematical Induction for positive integers (inference schema).
|- (x = y -> (ph <-> ps))   &   |- (x e. NN -> (A.y e. NN (y < x -> ps) -> ph))   =>   |- (x e. NN -> ph)
Theorem	uzinfm 6412	Extract the lower bound of a set of upper integers as its infimum. Note that the "`' <" argument turns supremum into infimum (for which we do not currently have a separate notation).
|- M e. ZZ   =>   |- sup((ZZ>` M), RR, `' < ) = M
Theorem	nninfm 6413	The infimum of the set of natural numbers is one.
|- sup(NN, RR, `' < ) = 1
Theorem	nn0infm 6414	The infimum of the set of nonnegative integers is zero. Note that "`' <" turns sup into inf.
|- sup(NN0, RR, `' < ) = 0
Theorem	infmssuzle 6415	The infimum of a subset of a set of upper integers is less than or equal to all members of the subset. Note that the "`' < " argument turns supremum into infimum (for which we do not currently have a separate notation).
|- ((S (_ (ZZ>` M) /\ -. S = (/) /\ A e. S) -> sup(S, RR, `' < ) <_ A)
Theorem	infmssuzcl 6416	The infimum of a subset of a set of upper integers belongs to the subset.
|- ((S (_ (ZZ>` M) /\ S =/= (/)) -> sup(S, RR, `' < ) e. S)
Finite intervals of integers
Syntax	cfz 6417	Extend class notation to include the notation for a contiguous finite set of integers. Read "M...N" as "the set of integers from M to N inclusive."
class ...
Definition	df-fz 6418	Define an operation that produces a finite set of sequential integers. Read "M...N" as "the set of integers from M to N inclusive." See fzvalt 6419 for its value and additional comments.
|- ... = {<.<.m, n>., z>. | ((m e. ZZ /\ n e. ZZ) /\ z = {k e. ZZ | (m <_ k /\ k <_ n)})}
Theorem	fzvalt 6419	The value of a finite set of sequential integers. E.g., 2...5 means the set {2, 3, 4, 5}. A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where NNk means our 1...k; he calls these sets segments of the integers.
|- ((M e. ZZ /\ N e. ZZ) -> (M...N) = {k e. ZZ | (M <_ k /\ k <_ N)})
Theorem	elfz1t 6420	Membership in a finite set of sequential integers.
|- ((M e. ZZ /\ N e. ZZ) -> (K e. (M...N) <-> (K e. ZZ /\ M <_ K /\ K <_ N)))
Theorem	elfzt 6421	Membership in a finite set of sequential integers.
|- ((K e. ZZ /\ M e. ZZ /\ N e. ZZ) -> (K e. (M...N) <-> (M <_ K /\ K <_ N)))
Theorem	elfz2t 6422	Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show M e. ZZ and N e. ZZ.
|- (N e. A -> (K e. (M...N) <-> ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) /\ (M <_ K /\ K <_ N))))
Theorem	elfzlem 6423	Lemma for elfzel1 6431 and others.
Theorem	elfz5t 6424	Membership in a finite set of sequential integers.
|- ((K e. (ZZ>` M) /\ N e. ZZ) -> (K e. (M...N) <-> K <_ N))
Theorem	elfz4t 6425	Membership in a finite set of sequential integers.
|- (((M e. ZZ /\ N e. ZZ /\ K e. ZZ) /\ (M <_ K /\ K <_ N)) -> K e. (M...N))
Theorem	elfzuzb 6426	Membership in a finite set of sequential integers in terms of sets of upper integers.
|- (N e. A -> (K e. (M...N) <-> (K e. (ZZ>` M) /\ N e. (ZZ>` K))))
Theorem	eluzfzt 6427	Membership in a finite set of sequential integers.
|- ((K e. (ZZ>` M) /\ N e. (ZZ>` K)) -> K e. (M...N))
Theorem	elfzuz3t 6428	Membership in a finite set of sequential integers implies membership in a set of upper integers.
|- ((N e. A /\ K e. (M...N)) -> N e. (ZZ>` K))
Theorem	elfzel2 6429	Membership in a finite set of sequential integer implies the upper bound is an integer.
|- N e. V   =>   |- (K e. (M...N) -> N e. ZZ)
Theorem	elfzel2g 6430	Membership in a finite set of sequential integers implies the upper bound is an integer.
|- ((N e. A /\ K e. (M...N)) -> N e. ZZ)
Theorem	elfzel1 6431	Membership in a finite set of sequential integer implies the lower bound is an integer.
|- (K e. (M...N) -> M e. ZZ)
Theorem	elfzelz 6432	A member of a finite set of sequential integer is an integer.
|- (K e. (M...N) -> K e. ZZ)
Theorem	elfzle1 6433	A member of a finite set of sequential integer is greater than or equal to the lower bound.
|- (K e. (M...N) -> M <_ K)
Theorem	elfzle2 6434	A member of a finite set of sequential integer is less than or equal to the upper bound.
|- (K e. (M...N) -> K <_ N)
Theorem	elfzle3 6435	Membership in a finite set of sequential integer implies the bounds are comparable.
|- ((N e. A /\ K e. (M...N)) -> M <_ N)
Theorem	elfzuz2t 6436	Implication of membership in a finite set of sequential integers.
|- ((N e. A /\ K e. (M...N)) -> N e. (ZZ>` M))
Theorem	eluzfz1t 6437	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>` M) -> M e. (M...N))
Theorem	elfzuzt 6438	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 6439	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>` M) -> N e. (M...N))
Theorem	eluzfz2b 6440	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>` M) <-> N e. (M...N))
Theorem	elfz3t 6441	Membership in a finite set of sequential integers containing one integer.
|- (N e. ZZ -> N e. (N...N))
Theorem	elfz1eqt 6442	Membership in a finite set of sequential integers containing one integer.
|- (K e. (N...N) -> K = N)
Theorem	fznt 6443	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 6444	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 6445	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 6446	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 6447	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 6448	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 6449	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 6450	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 6451	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 6452	A finite set of sequential integers can represent an ordered pair.
|- ((N e. (ZZ>`