mmtheorems64 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6301-6400 - Page 64 of 123

Type	Label	Description
Statement
Theorem	nn0addge2i 6301	A number is less than or equal to itself plus a nonnegative integer.
|- A e. RR   &   |- N e. NN0   =>   |- A <_ (N + A)
Theorem	nn0le2xi 6302	A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
|- N e. NN0   =>   |- N <_ (2 x. N)
Theorem	nn0lele2xi 6303	'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
|- M e. NN0   &   |- N e. NN0   =>   |- (N <_ M -> N <_ (2 x. M))
Integers (as a subset of complex numbers)
Definition	df-z 6304	Define the set of integers. Definition of integers in [Apostol] p. 22.
|- ZZ = {n e. RR | (n = 0 \/ n e. NN \/ -un e. NN)}
Theorem	elz 6305	Membership in the set of integers.
|- (N e. ZZ <-> (N e. RR /\ (N = 0 \/ N e. NN \/ -uN e. NN)))
Theorem	nnnegz 6306	The negative of a natural number is an integer.
|- (N e. NN -> -uN e. ZZ)
Theorem	zre 6307	An integer is a real.
|- (N e. ZZ -> N e. RR)
Theorem	zcn 6308	An integer is a complex number.
|- (N e. ZZ -> N e. CC)
Theorem	zrei 6309	An integer is a real number.
|- A e. ZZ   =>   |- A e. RR
Theorem	zssre 6310	The integers are a subset of the reals.
|- ZZ (_ RR
Theorem	zsscn 6311	The integers are a subset of the complex numbers.
|- ZZ (_ CC
Theorem	zex 6312	The set of integers exists.
|- ZZ e. V
Theorem	elnnz 6313	Natural number property expressed in terms of integers.
|- (N e. NN <-> (N e. ZZ /\ 0 < N))
Theorem	0z 6314	Zero is an integer.
|- 0 e. ZZ
Theorem	elnn0z 6315	Nonnegative integer property expressed in terms of integers.
|- (N e. NN0 <-> (N e. ZZ /\ 0 <_ N))
Theorem	elznn0nn 6316	Integer property expressed in terms nonnegative integers and natural numbers.
|- (N e. ZZ <-> (N e. NN0 \/ (N e. RR /\ -uN e. NN)))
Theorem	elznn0 6317	Integer property expressed in terms of nonnegative integers.
|- (N e. ZZ <-> (N e. RR /\ (N e. NN0 \/ -uN e. NN0)))
Theorem	elznn 6318	Integer property expressed in terms natural numbers and nonnegative integers.
|- (N e. ZZ <-> (N e. RR /\ (N e. NN \/ -uN e. NN0)))
Theorem	nnssz 6319	Natural numbers are a subset of integers.
|- NN (_ ZZ
Theorem	nn0ssz 6320	Nonnegative integers are a subset of the integers.
|- NN0 (_ ZZ
Theorem	nnz 6321	A natural number is an integer.
|- (N e. NN -> N e. ZZ)
Theorem	nn0z 6322	A nonnegative integer is an integer.
|- (N e. NN0 -> N e. ZZ)
Theorem	elnnz1 6323	Natural number property expressed in terms of integers.
|- (N e. NN <-> (N e. ZZ /\ 1 <_ N))
Theorem	znnnlt1 6324	An integer is not a natural number iff it is less than one.
|- (N e. ZZ -> (-. N e. NN <-> N < 1))
Theorem	nnzrab 6325	Natural numbers expressed as a subset of integers.
|- NN = {x e. ZZ | 1 <_ x}
Theorem	nn0zrab 6326	Nonnegative integers expressed as a subset of integers.
|- NN0 = {x e. ZZ | 0 <_ x}
Theorem	1z 6327	One is an integer.
|- 1 e. ZZ
Theorem	2z 6328	Two is an integer.
|- 2 e. ZZ
Theorem	nn0sub 6329	Subtraction of nonnegative integers.
|- ((M e. NN0 /\ N e. NN0) -> (M <_ N <-> (N - M) e. NN0))
Theorem	nn0sub2 6330	Subtraction of nonnegative integers.
|- ((M e. NN0 /\ N e. NN0 /\ M <_ N) -> (N - M) e. NN0)
Theorem	znegcl 6331	Closure law for negative integers.
|- (N e. ZZ -> -uN e. ZZ)
Theorem	nn0negz 6332	The negative of a nonnegative integer is an integer.
|- (N e. NN0 -> -uN e. ZZ)
Theorem	zaddcl 6333	Closure of addition of integers.
|- ((M e. ZZ /\ N e. ZZ) -> (M + N) e. ZZ)
Theorem	peano2z 6334	Second Peano postulate generalized to integers.
|- (N e. ZZ -> (N + 1) e. ZZ)
Theorem	peano2zdi 6335	Deduction from second Peano postulate generalized to integers.
|- (ph -> N e. ZZ)   =>   |- (ph -> (N + 1) e. ZZ)
Theorem	zsubcl 6336	Closure of subtraction of integers.
|- ((M e. ZZ /\ N e. ZZ) -> (M - N) e. ZZ)
Theorem	peano2zm 6337	"Reverse" second Peano postulate for integers.
|- (N e. ZZ -> (N - 1) e. ZZ)
Theorem	zrevaddcl 6338	Reverse closure law for addition of integers.
|- (N e. ZZ -> ((M e. CC /\ (M + N) e. ZZ) <-> M e. ZZ))
Theorem	elnn0nn 6339	The nonnegative integer property expressed in terms of natural numbers.
|- (N e. NN0 <-> (N e. CC /\ (N + 1) e. NN))
Theorem	elnnnn0 6340	The natural number property expressed in terms of nonnegative integers.
|- (N e. NN <-> (N e. CC /\ (N - 1) e. NN0))
Theorem	elnnnn0b 6341	The natural number property expressed in terms of nonnegative integers.
|- (N e. NN <-> (N e. NN0 /\ 0 < N))
Theorem	elnnnn0c 6342	The natural number property expressed in terms of nonnegative integers.
|- (N e. NN <-> (N e. NN0 /\ 1 <_ N))
Theorem	nn0p1nn 6343	A nonnegative integer plus 1 is a natural number. (Contributed by Raph Levien, 30-Jun-2006.)
|- (N e. NN0 -> (N + 1) e. NN)
Theorem	nnm1nn0 6344	A natural number minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.)
|- (N e. NN -> (N - 1) e. NN0)
Theorem	znnsub 6345	The positive difference of unequal integers is a natural number. (Generalization of nnsub 6103.)
|- ((M e. ZZ /\ N e. ZZ) -> (M < N <-> (N - M) e. NN))
Theorem	znn0sub 6346	The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 6329.)
|- ((M e. ZZ /\ N e. ZZ) -> (M <_ N <-> (N - M) e. NN0))
Theorem	znn0sub2 6347	The nonnegative difference of integers is a nonnegative integer.
|- ((M e. ZZ /\ N e. ZZ /\ M <_ N) -> (N - M) e. NN0)
Theorem	zmulcl 6348	Closure of multiplication of integers.
|- ((M e. ZZ /\ N e. ZZ) -> (M x. N) e. ZZ)
Theorem	zltp1le 6349	Integer ordering relation.
|- ((M e. ZZ /\ N e. ZZ) -> (M < N <-> (M + 1) <_ N))
Theorem	zleltp1 6350	Integer ordering relation.
|- ((M e. ZZ /\ N e. ZZ) -> (M <_ N <-> M < (N + 1)))
Theorem	zlem1lt 6351	Integer ordering relation.
|- ((M e. ZZ /\ N e. ZZ) -> (M <_ N <-> (M - 1) < N))
Theorem	zltlem1 6352	Integer ordering relation.
|- ((M e. ZZ /\ N e. ZZ) -> (M < N <-> M <_ (N - 1)))
Theorem	nn0lem1lt 6353	Nonnegative integer ordering relation.
|- ((M e. NN0 /\ N e. NN0) -> (M <_ N <-> (M - 1) < N))
Theorem	nnlem1lt 6354	Natural number ordering relation.
|- ((M e. NN /\ N e. NN) -> (M <_ N <-> (M - 1) < N))
Theorem	nnltlem1 6355	Natural number ordering relation.
|- ((M e. NN /\ N e. NN) -> (M < N <-> M <_ (N - 1)))
Theorem	zdiv 6356	Two ways to express "M divides N.
|- ((M e. NN /\ N e. ZZ) -> (E.k e. ZZ (M x. k) = N <-> (N / M) e. ZZ))
Theorem	zdivadd 6357	Property of divisibility: if D divides A and B then it divides A + B.
|- (((D e. NN /\ A e. ZZ /\ B e. ZZ) /\ ((A / D) e. ZZ /\ (B / D) e. ZZ)) -> ((A + B) / D) e. ZZ)
Theorem	zdivmul 6358	Property of divisibility: if D divides A then it divides B x. A.
|- (((D e. NN /\ A e. ZZ /\ B e. ZZ) /\ (A / D) e. ZZ) -> ((B x. A) / D) e. ZZ)
Theorem	z2ge 6359	There exists an integer greater than or equal to any two others.
|- ((M e. ZZ /\ N e. ZZ) -> E.k e. ZZ (M <_ k /\ N <_ k))
Theorem	zextle 6360	An extensionality-like property for integer ordering.
|- ((M e. ZZ /\ N e. ZZ /\ A.k e. ZZ (k <_ M <-> k <_ N)) -> M = N)
Theorem	zextlt 6361	An extensionality-like property for integer ordering.
|- ((M e. ZZ /\ N e. ZZ /\ A.k e. ZZ (k < M <-> k < N)) -> M = N)
Theorem	recnz 6362	The reciprocal of a number greater than 1 is not an integer.
|- ((A e. RR /\ 1 < A) -> -. (1 / A) e. ZZ)
Theorem	btwnnz 6363	A number between an integer and its successor is not an integer.
|- ((A e. ZZ /\ A < B /\ B < (A + 1)) -> -. B e. ZZ)
Theorem	gtndiv 6364	A larger number does not divide a smaller natural number.
|- ((A e. RR /\ B e. NN /\ B < A) -> -. (B / A) e. ZZ)
Theorem	halfnz 6365	One-half is not an integer.
|- -. (1 / 2) e. ZZ
Theorem	prime 6366	Two ways to express "A is a prime number (or 1)."
|- (A e. NN -> (A.x e. NN ((A / x) e. NN -> (x = 1 \/ x = A)) <-> A.x e. NN ((1 < x /\ x <_ A /\ (A / x) e. NN) -> x = A)))
Theorem	msqznn 6367	The square of a non-zero integer is a natural number.
|- ((A e. ZZ /\ A =/= 0) -> (A x. A) e. NN)
Theorem	nneoi 6368	A natural number is even or odd but not both.
|- N e. NN   =>   |- ((N / 2) e. NN <-> -. ((N + 1) / 2) e. NN)
Theorem	nneo 6369	A natural number is even or odd but not both.
|- (N e. NN -> ((N / 2) e. NN <-> -. ((N + 1) / 2) e. NN))
Theorem	zeo 6370	An integer is even or odd.
|- (N e. ZZ -> ((N / 2) e. ZZ \/ ((N + 1) / 2) e. ZZ))
Theorem	zneo 6371	No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28.
|- ((A e. ZZ /\ B e. ZZ) -> (2 x. A) =/= ((2 x. B) + 1))
Theorem	peano2uz2 6372	Second Peano postulate for upper integers.
|- ((A e. ZZ /\ B e. {x e. ZZ | A <_ x}) -> (B + 1) e. {x e. ZZ | A <_ x})
Theorem	dfuzi 6373	An expression for the upper integers that start at N that is analogous to df-n 6070 for natural numbers. Warning: The HTML proof page is 1/2 megabyte in size.
|- N e. ZZ   =>   |- {z e. ZZ | N <_ z} = |^|{x | (N e. x /\ A.y e. x (y + 1) e. x)}
Theorem	peano5uzi 6374	Peano's inductive postulate for upper integers.
|- A e. V   &   |- N e. ZZ   =>   |- ((N e. A /\ A.x e. A (x + 1) e. A) -> {k e. ZZ | N <_ k} (_ A)
Theorem	peano5uzti 6375	Peano's inductive postulate for upper integers.
|- A e. V   =>   |- (N e. ZZ -> ((N e. A /\ A.x e. A (x + 1) e. A) -> {k e. ZZ | N <_ k} (_ A))
Theorem	uzind 6376	Induction on the upper integers that start at M. The first four hypotheses give us the substitution instances we need; 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)   &   |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (ch -> th))   =>   |- ((M e. ZZ /\ N e. ZZ /\ M <_ N) -> ta)
Theorem	uzind2 6377	Induction on the upper integers that start after an integer M. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis.
|- (j = (M + 1) -> (ph <-> ps))   &   |- (j = k -> (ph <-> ch))   &   |- (j = (k + 1) -> (ph <-> th))   &   |- (j = N -> (ph <-> ta))   &   |- (M e. ZZ -> ps)   &   |- ((M e. ZZ /\ k e. ZZ /\ M < k) -> (ch -> th))   =>   |- ((M e. ZZ /\ N e. ZZ /\ M < N) -> ta)
Theorem	uzind3 6378	Induction on the upper integers that start 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 = m -> (ph <-> ch))   &   |- (j = (m + 1) -> (ph <-> th))   &   |- (j = N -> (ph <-> ta))   &   |- (M e. ZZ -> ps)   &   |- ((M e. ZZ /\ m e. {k e. ZZ | M <_ k}) -> (ch -> th))   =>   |- ((M e. ZZ /\ N e. {k e. ZZ | M <_ k}) -> ta)
Theorem	uzindOLD 6379	Induction on the upper integers that start at an integer B. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. Warning: The HTML proof page is 3/4 megabyte in size. An attempt to shorten it is on my to-do list.
|- (x = B -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = (y + 1) -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   &   |- ps   &   |- (((y e. ZZ /\ B e. ZZ) /\ B <_ y) -> (ch -> th))   =>   |- (((A e. ZZ /\ B e. ZZ) /\ B <_ A) -> ta)
Theorem	uzind3OLD 6380	Induction on the set of upper integers that starts at B. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction hypothesis.
|- (x = B -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = (y + 1) -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   &   |- ps   &   |- ((B e. ZZ /\ y e. {z e. ZZ | B <_ z}) -> (ch -> th))   =>   |- ((B e. ZZ /\ A e. {z e. ZZ | B <_ z}) -> ta)
Theorem	uzwo4OLD 6381	Well-ordering principle: any non-empty subset of the upper integers has a least element.
|- ((B e. ZZ /\ (A (_ {z e. ZZ | B <_ z} /\ A =/= (/))) -> E.x e. A A.y e. A x <_ y)
Theorem	uzwo5OLD 6382	Well-ordering principle: any non-empty subset of upper integers has a unique least element.
|- ((B e. ZZ /\ (A (_ {z e. ZZ | B <_ z} /\ A =/= (/))) -> E!x e. A A.y e. A x <_ y)
Theorem	nn0ind 6383	Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis.
|- (x = 0 -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = (y + 1) -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   &   |- ps   &   |- (y e. NN0 -> (ch -> th))   =>   |- (A e. NN0 -> ta)
Theorem	nn0indALT 6384	Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis.
|- (y e. NN0 -> (ch -> th))   &   |- ps   &   |- (x = 0 -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = (y + 1) -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   =>   |- (A e. NN0 -> ta)
Theorem	nn0ind-raph 6385	Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by Raph Levien, 10-Apr-2004. Raph says: "This seems a bit painful. I wonder if an explicit substitution version would be easier.")
|- (x = 0 -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = (y + 1) -> (ph <-> th))   &   |- (x = A -> (ph <-> ta))   &   |- ps   &   |- (y e. NN0 -> (ch -> th))   =>   |- (A e. NN0 -> ta)
Theorem	zindd 6386	Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.)
|- (x = 0 -> (ph <-> ps))   &   |- (x = y -> (ph <-> ch))   &   |- (x = (y + 1) -> (ph <-> ta))   &   |- (x = -uy -> (ph <-> th))   &   |- (x = A -> (ph <-> et))   &   |- (ze -> ps)   &   |- (ze -> (y e. NN0 -> (ch -> ta)))   &   |- (ze -> (y e. NN -> (ch -> th)))   =>   |- (ze -> (A e. ZZ -> et))
Theorem	btwnz 6387	Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28.
|- (A e. RR -> (E.x e. ZZ x < A /\ E.y e. ZZ A < y))
Well-ordering principle for bounded-below sets of integers
Theorem	uzwo3lem1 6388	Lemma for uzwo3 6390 and zmin 6391.
Theorem	uzwo3lem2 6389	Lemma for uzwo3 6390.
Theorem	uzwo3 6390	Well-ordering principle: any non-empty subset of upper integers has a unique least element. This generalization of uzwo2 6584 allows the lower bound B to be any real number. See also nnwo 6585 and nnwos 6587.
|- ((B e. RR /\ (A (_ {z e. ZZ | B <_ z} /\ A =/= (/))) -> E!x e. A A.y e. A x <_ y)
Theorem	zmin 6391	There is a unique smallest integer greater than or equal to a given real number.
|- (A e. RR -> E!x e. ZZ (A <_ x /\ A.y e. ZZ (A <_ y -> x <_ y)))
Theorem	zmax 6392	There is a unique largest integer less than or equal to a given real number.
|- (A e. RR -> E!x e. ZZ (x <_ A /\ A.y e. ZZ (y <_ A -> y <_ x)))
Theorem	zbtwnre 6393	There is a unique integer between a real number and the number plus one. Exercise 5 of [Apostol] p. 28.
|- (A e. RR -> E!x e. ZZ (A <_ x /\ x < (A + 1)))
Theorem	rebtwnz 6394	There is a unique greatest integer less than or equal to a real number. Exercise 4 of [Apostol] p. 28.
|- (A e. RR -> E!x e. ZZ (x <_ A /\ A < (x + 1)))
Rational numbers (as a subset of complex numbers)
Definition	df-q 6395	Define the set of rational numbers. Definition of rationals in [Apostol] p. 22.
|- QQ = {x | E.m e. ZZ E.n e. NN x = (m / n)}
Theorem	elq 6396	Membership in the set of rationals.
|- (A e. QQ <-> E.x e. ZZ E.y e. NN A = (x / y))
Theorem	znq 6397	The ratio of an integer and a natural number is a rational number.
|- ((A e. ZZ /\ B e. NN) -> (A / B) e. QQ)
Theorem	qre 6398	A rational number is a real number.
|- (A e. QQ -> A e. RR)
Theorem	zq 6399	An integer is a rational number.
|- (A e. ZZ -> A e. QQ)
Theorem	zssq 6400	The integers are a subset of the rationals.
|- ZZ (_ QQ