mmtheorems77 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7601-7700 - Page 77 of 175

Type	Label	Description
Statement
Theorem	uz11 7601	The upper integers function is one-to-one.
|- (M e. ZZ -> ((ZZ>=` M) = (ZZ>=` N) <-> M = N))
Theorem	eluzp1m1 7602	Membership in the next set of upper integers.
|- ((M e. ZZ /\ N e. (ZZ>=` (M + 1))) -> (N - 1) e. (ZZ>=` M))
Theorem	eluzp1l 7603	Strict ordering implied by membership in the next set of upper integers.
|- ((M e. ZZ /\ N e. (ZZ>=` (M + 1))) -> M < N)
Theorem	eluzp1p1 7604	Membership in the next set of upper integers.
|- (N e. (ZZ>=` M) -> (N + 1) e. (ZZ>=` (M + 1)))
Theorem	eluzaddi 7605	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 7606	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 7607	Nonnegative integers expressed as a set of upper integers.
|- NN0 = (ZZ>=` 0)
Theorem	nnuz 7608	Natural numbers expressed as a set of upper integers.
|- NN = (ZZ>=` 1)
Theorem	elnnuz 7609	A natural number expressed as a member of a set of upper integers.
|- (N e. NN <-> N e. (ZZ>=` 1))
Theorem	elnn0uz 7610	A nonnegative integer expressed as a member a set of upper integers.
|- (N e. NN0 <-> N e. (ZZ>=` 0))
Theorem	raluz 7611	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 7612	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 7613	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 7614	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 7615	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 7616	Second Peano postulate for a set of upper integers.
|- (N e. (ZZ>=` M) -> (N + 1) e. (ZZ>=` M))
Theorem	peano2uzr 7617	Reversed second Peano axiom for upper integers.
|- ((M e. ZZ /\ N e. (ZZ>=` (M + 1))) -> N e. (ZZ>=` M))
Theorem	uzaddcl 7618	Addition closure law for a set of upper integers.
|- ((N e. (ZZ>=` M) /\ K e. NN0) -> (N + K) e. (ZZ>=` M))
Theorem	uzind4 7619	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 7620	Alternate version of uzind4 7619 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 7619- 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 7621	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 7622	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 7621 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 7623	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 7624	Well-ordering principle: any non-empty subset of a set of upper integers has a least element.
|- ((S C_ (ZZ>=` M) /\ S =/= (/)) -> E.j e. S A.k e. S j <_ k)
Theorem	uzwoOLD 7625	Well-ordering principle: any non-empty subset of the upper integers has a least element.
|- ((S C_ (ZZ>=` M) /\ -. S = (/)) -> E.j e. S A.k e. S j <_ k)
Theorem	uzwo2 7626	Well-ordering principle: any non-empty subset of upper integers has a unique least element.
|- ((S C_ (ZZ>=` M) /\ S =/= (/)) -> E!j e. S A.k e. S j <_ k)
Theorem	nnwo 7627	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 C_ NN /\ A =/= (/)) -> E.x e. A A.y e. A x <_ y)
Theorem	nnwof 7628	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 C_ NN /\ A =/= (/)) -> E.x e. A A.y e. A x <_ y)
Theorem	nnwos 7629	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 7630	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	uzinfmi 7631	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 7632	The infimum of the set of natural numbers is one. Note that "`' < " turns sup into inf.
|- sup(NN, RR, `' < ) = 1
Theorem	nn0infm 7633	The infimum of the set of nonnegative integers is zero. Note that "`' < " turns sup into inf.
|- sup(NN0, RR, `' < ) = 0
Theorem	infmssuzle 7634	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 C_ (ZZ>=` M) /\ A e. S) -> sup(S, RR, `' < ) <_ A)
Theorem	infmssuzleOLD 7635	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 C_ (ZZ>=` M) /\ S =/= (/) /\ A e. S) -> sup(S, RR, `' < ) <_ A)
Theorem	infmssuzcl 7636	The infimum of a subset of a set of upper integers belongs to the subset.
|- ((S C_ (ZZ>=` M) /\ S =/= (/)) -> sup(S, RR, `' < ) e. S)
Finite intervals of integers
Syntax	cfz 7637	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 7638	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 fzval 7639 for its value and additional comments.
|- ... = {<.<.m, n>., z>. | ((m e. ZZ /\ n e. ZZ) /\ z = {k e. ZZ | (m <_ k /\ k <_ n)})}
Theorem	fzval 7639	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 NN_k 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	elfz1 7640	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	elfz 7641	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	elfz2 7642	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 7643	Lemma for elfzel1 7651 and others.
Theorem	elfz5 7644	Membership in a finite set of sequential integers.
|- ((K e. (ZZ>=` M) /\ N e. ZZ) -> (K e. (M...N) <-> K <_ N))
Theorem	elfz4 7645	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 7646	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	eluzfz 7647	Membership in a finite set of sequential integers.
|- ((K e. (ZZ>=` M) /\ N e. (ZZ>=` K)) -> K e. (M...N))
Theorem	elfzuz3 7648	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	elfzel2i 7649	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 7650	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 7651	Membership in a finite set of sequential integer implies the lower bound is an integer.
|- (K e. (M...N) -> M e. ZZ)
Theorem	elfzelz 7652	A member of a finite set of sequential integer is an integer.
|- (K e. (M...N) -> K e. ZZ)
Theorem	elfzle1 7653	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 7654	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 7655	Membership in a finite set of sequential integer implies the bounds are comparable.
|- ((N e. A /\ K e. (M...N)) -> M <_ N)
Theorem	elfzuz2 7656	Implication of membership in a finite set of sequential integers.
|- ((N e. A /\ K e. (M...N)) -> N e. (ZZ>=` M))
Theorem	eluzfz1 7657	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>=` M) -> M e. (M...N))
Theorem	elfzuz 7658	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	eluzfz2 7659	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>=` M) -> N e. (M...N))
Theorem	eluzfz2b 7660	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>=` M) <-> N e. (M...N))
Theorem	elfz3 7661	Membership in a finite set of sequential integers containing one integer.
|- (N e. ZZ -> N e. (N...N))
Theorem	elfz1eq 7662	Membership in a finite set of sequential integers containing one integer.
|- (K e. (N...N) -> K = N)
Theorem	fzn 7663	A finite set of sequential integers is empty if the bounds are reversed.
|- ((M e. ZZ /\ N e. ZZ) -> (N < M <-> (M...N) = (/)))
Theorem	fzen 7664	A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)
|- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (M...N) ~~ ((M + K)...(N + K)))
Theorem	fz01en 7665	0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)
|- (N e. ZZ -> (0...(N - 1)) ~~ (1...N))
Theorem	elfznn 7666	A member of a finite set of sequential integers starting at 1 is a natural number.
|- (K e. (1...N) -> K e. NN)
Theorem	elfz2nn0 7667	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	elfznn0 7668	A member of a finite set of sequential integers starting at 0 is a nonnegative integer.
|- (K e. (0...N) -> K e. NN0)
Theorem	elfz3nn0 7669	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	fznn0sub 7670	Subtraction closure for a member of a finite set of sequential integers.
|- ((N e. A /\ K e. (M...N)) -> (N - K) e. NN0)
Theorem	fznn0sub2 7671	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	fzaddel 7672	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	fzsubel 7673	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	fzopth 7674	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	fzss1 7675	Subset relationship for finite sets of sequential integers.
|- ((K e. (ZZ>=` M) /\ N e. ZZ) -> (K...N) C_ (M...N))
Theorem	fzss2 7676	Subset relationship for finite sets of sequential integers.
|- ((N e. (ZZ>=` K) /\ M e. ZZ) -> (M...K) C_ (M...N))
Theorem	fzssuz 7677	A finite set of sequential integers is a subset of a set of upper integers.
|- (M...N) C_ (ZZ>=` M)
Theorem	fzsuc 7678	Join a successor to the end of a finite set of sequential integers.
|- ((M e. ZZ /\ N e. ZZ /\ M <_ (N + 1)) -> (M...(N + 1)) = ((M...N) u. {(N + 1)}))
Theorem	fzssp1 7679	Subset relationship for finite sets of sequential integers.
|- ((M e. ZZ /\ N e. ZZ) -> (M...N) C_ (M...(N + 1)))
Theorem	fzp1ss 7680	Subset relationship for finite sets of sequential integers.
|- ((M e. ZZ /\ N e. ZZ) -> ((M + 1)...N) C_ (M...N))
Theorem	fzelp1 7681	Membership in a set of sequential integers with an appended element.
|- ((N e. A /\ K e. (M...N)) -> K e. (M...(N + 1)))
Theorem	fzelp1i 7682	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 7683	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	fzsn 7684	A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009 )
|- (M e. ZZ -> (M...M) = {M})
Theorem	fzpr 7685	A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009 )
|- (M e. ZZ -> (M...(M + 1)) = {M, (M + 1)})
Theorem	fztp 7686	A finite interval of integers with three elements.
|- (M e. ZZ -> (M...(M + 2)) = {M, (M + 1), (M + 2)})
Theorem	fzprval 7687	Two ways of defining the first two values of a sequence on NN.
|- (A.x e. (1...2)(F` x) = if(x = 1, A, B) <-> ((F` 1) = A /\ (F` 2) = B))
Theorem	fztpval 7688	Two ways of defining the first three values of a sequence on NN.
|- (A.x e. (1...3)(F` x) = if(x = 1, A, if(x = 2, B, C)) <-> ((F` 1) = A /\ (F` 2) = B /\ (F` 3) = C))
Theorem	fzrev 7689	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	fzrev2 7690	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	fzrev2i 7691	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	fzrev3 7692	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	fzrev3i 7693	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	fznn0 7694	Finite set of sequential integers starting at 0.
|- (N e. NN0 -> (K e. (0...N) <-> (K e. NN0 /\ K <_ N)))
Theorem	fznn 7695	Finite set of sequential integers starting at 1.
|- (N e. NN -> (K e. (1...N) <-> (K e. NN /\ K <_ N)))
Theorem	fz1sbc 7696	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	fzneuz 7697	No finite set of sequential integers equals a set of upper integers.
|- ((N e. (ZZ>=` M) /\ K e. ZZ) -> -. (M...N) = (ZZ>=` K))
Theorem	fzrevral 7698	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	fzrevral2 7699	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	fzrevral3 7700	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))