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	uzind4 6401	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 6402	Alternate version of uzind4 6401 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 6401 - 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 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.
|- (M e. ZZ -> [M / k]ph)   &   |- (k e. (ZZ>` M) -> (ph -> [(k + 1) / k]ph))   =>   |- (N e. (ZZ>` M) -> [N / k]ph)
Theorem	uzind4s2 6404	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 6403 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 6405	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 6406	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 6407	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 6408	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 6409	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 6410	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 6411	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 6412	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 6413	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 6414	The infimum of the set of natural numbers is one.
|- sup(NN, RR, `' < ) = 1
Theorem	nn0infm 6415	The infimum of the set of nonnegative integers is zero. Note that "`' <" turns sup into inf.
|- sup(NN0, RR, `' < ) = 0
Theorem	infmssuzle 6416	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 6417	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 6418	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 6419	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 6420 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 6420	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 6421	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 6422	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 6423	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 6424	Lemma for elfzel1 6432 and others.
Theorem	elfz5t 6425	Membership in a finite set of sequential integers.
|- ((K e. (ZZ>` M) /\ N e. ZZ) -> (K e. (M...N) <-> K <_ N))
Theorem	elfz4t 6426	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 6427	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 6428	Membership in a finite set of sequential integers.
|- ((K e. (ZZ>` M) /\ N e. (ZZ>` K)) -> K e. (M...N))
Theorem	elfzuz3t 6429	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 6430	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 6431	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 6432	Membership in a finite set of sequential integer implies the lower bound is an integer.
|- (K e. (M...N) -> M e. ZZ)
Theorem	elfzelz 6433	A member of a finite set of sequential integer is an integer.
|- (K e. (M...N) -> K e. ZZ)
Theorem	elfzle1 6434	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 6435	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 6436	Membership in a finite set of sequential integer implies the bounds are comparable.
|- ((N e. A /\ K e. (M...N)) -> M <_ N)
Theorem	elfzuz2t 6437	Implication of membership in a finite set of sequential integers.
|- ((N e. A /\ K e. (M...N)) -> N e. (ZZ>` M))
Theorem	eluzfz1t 6438	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>` M) -> M e. (M...N))
Theorem	elfzuzt 6439	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 6440	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>` M) -> N e. (M...N))
Theorem	eluzfz2b 6441	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>` M) <-> N e. (M...N))
Theorem	elfz3t 6442	Membership in a finite set of sequential integers containing one integer.
|- (N e. ZZ -> N e. (N...N))
Theorem	elfz1eqt 6443	Membership in a finite set of sequential integers containing one integer.
|- (K e. (N...N) -> K = N)
Theorem	fznt 6444	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 6445	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 6446	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 6447	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 6448	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 6449	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 6450	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 6451	Membership of a sum in a finite set of sequential integers.
|- (((M e. ZZ /\ N e. ZZ) /\ (J e. ZZ /\ K