mmtheorems62 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6101-6200 - Page 62 of 107

Type	Label	Description
Statement
Theorem	nn0lele2x 6101	'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 6102	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 6103	Membership in the set of integers.
|- (N e. ZZ <-> (N e. RR /\ (N = 0 \/ N e. NN \/ -uN e. NN)))
Theorem	nnnegz 6104	The negative of a natural number is an integer.
|- (N e. NN -> -uN e. ZZ)
Theorem	zret 6105	An integer is a real.
|- (N e. ZZ -> N e. RR)
Theorem	zcnt 6106	An integer is a complex number.
|- (N e. ZZ -> N e. CC)
Theorem	zre 6107	An integer is a real number.
|- A e. ZZ   =>   |- A e. RR
Theorem	zssre 6108	The integers are a subset of the reals.
|- ZZ (_ RR
Theorem	zsscn 6109	The integers are a subset of the complex numbers.
|- ZZ (_ CC
Theorem	zex 6110	The set of integers exists.
|- ZZ e. V
Theorem	elnnz 6111	Natural number property expressed in terms of integers.
|- (N e. NN <-> (N e. ZZ /\ 0 < N))
Theorem	0z 6112	Zero is an integer.
|- 0 e. ZZ
Theorem	elnn0z 6113	Nonnegative integer property expressed in terms of integers.
|- (N e. NN0 <-> (N e. ZZ /\ 0 <_ N))
Theorem	elznn0nn 6114	Integer property expressed in terms nonnegative integers and natural numbers.
|- (N e. ZZ <-> (N e. NN0 \/ (N e. RR /\ -uN e. NN)))
Theorem	elznn0 6115	Integer property expressed in terms of nonnegative integers.
|- (N e. ZZ <-> (N e. RR /\ (N e. NN0 \/ -uN e. NN0)))
Theorem	elznn 6116	Integer property expressed in terms natural numbers and nonnegative integers.
|- (N e. ZZ <-> (N e. RR /\ (N e. NN \/ -uN e. NN0)))
Theorem	nnssz 6117	Natural numbers are a subset of integers.
|- NN (_ ZZ
Theorem	nn0ssz 6118	Nonnegative integers are a subset of the integers.
|- NN0 (_ ZZ
Theorem	nnzt 6119	A natural number is an integer.
|- (N e. NN -> N e. ZZ)
Theorem	nn0zt 6120	A nonnegative integer is an integer.
|- (N e. NN0 -> N e. ZZ)
Theorem	elnnz1 6121	Natural number property expressed in terms of integers.
|- (N e. NN <-> (N e. ZZ /\ 1 <_ N))
Theorem	znnnlt1t 6122	An integer is not a natural number iff it is less than one.
|- (N e. ZZ -> (-. N e. NN <-> N < 1))
Theorem	nnzrab 6123	Natural numbers expressed as a subset of integers.
|- NN = {x e. ZZ | 1 <_ x}
Theorem	nn0zrab 6124	Nonnegative integers expressed as a subset of integers.
|- NN0 = {x e. ZZ | 0 <_ x}
Theorem	1z 6125	One is an integer.
|- 1 e. ZZ
Theorem	2z 6126	Two is an integer.
|- 2 e. ZZ
Theorem	nn0subt 6127	Subtraction of nonnegative integers.
|- ((M e. NN0 /\ N e. NN0) -> (M <_ N <-> (N - M) e. NN0))
Theorem	nn0sub2t 6128	Subtraction of nonnegative integers.
|- ((M e. NN0 /\ N e. NN0 /\ M <_ N) -> (N - M) e. NN0)
Theorem	znegclt 6129	Closure law for negative integers.
|- (N e. ZZ -> -uN e. ZZ)
Theorem	nn0negz 6130	The negative of a nonnegative integer is an integer.
|- (N e. NN0 -> -uN e. ZZ)
Theorem	zaddclt 6131	Closure of addition of integers.
|- ((M e. ZZ /\ N e. ZZ) -> (M + N) e. ZZ)
Theorem	peano2z 6132	Second Peano postulate generalized to integers.
|- (N e. ZZ -> (N + 1) e. ZZ)
Theorem	peano2zd 6133	Deduction from second Peano postulate generalized to integers.
|- (ph -> N e. ZZ)   =>   |- (ph -> (N + 1) e. ZZ)
Theorem	zsubclt 6134	Closure of subtraction of integers.
|- ((M e. ZZ /\ N e. ZZ) -> (M - N) e. ZZ)
Theorem	peano2zm 6135	"Reverse" second Peano postulate for integers.
|- (N e. ZZ -> (N - 1) e. ZZ)
Theorem	zrevaddclt 6136	Reverse closure law for addition of integers.
|- (N e. ZZ -> ((M e. CC /\ (M + N) e. ZZ) <-> M e. ZZ))
Theorem	elnn0nn 6137	The nonnegative integer property expressed in terms of natural numbers.
|- (N e. NN0 <-> (N e. CC /\ (N + 1) e. NN))
Theorem	elnnnn0 6138	The natural number property expressed in terms of nonnegative integers.
|- (N e. NN <-> (N e. CC /\ (N - 1) e. NN0))
Theorem	elnnnn0b 6139	The natural number property expressed in terms of nonnegative integers.
|- (N e. NN <-> (N e. NN0 /\ 0 < N))
Theorem	elnnnn0c 6140	The natural number property expressed in terms of nonnegative integers.
|- (N e. NN <-> (N e. NN0 /\ 1 <_ N))
Theorem	nn0p1nnt 6141	A nonnegative integer plus 1 is a natural number. (Contributed by Raph Levien, 30-Jun-2006.)
|- (N e. NN0 -> (N + 1) e. NN)
Theorem	nnm1nn0t 6142	A natural number minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.)
|- (N e. NN -> (N - 1) e. NN0)
Theorem	znnsubt 6143	The positive difference of unequal integers is a natural number. (Generalization of nnsubt 5923.)
|- ((M e. ZZ /\ N e. ZZ) -> (M < N <-> (N - M) e. NN))
Theorem	znn0subt 6144	The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0subt 6127.)
|- ((M e. ZZ /\ N e. ZZ) -> (M <_ N <-> (N - M) e. NN0))
Theorem	znn0sub2t 6145	The nonnegative difference of integers is a nonnegative integer.
|- ((M e. ZZ /\ N e. ZZ /\ M <_ N) -> (N - M) e. NN0)
Theorem	zmulclt 6146	Closure of multiplication of integers.
|- ((M e. ZZ /\ N e. ZZ) -> (M x. N) e. ZZ)
Theorem	zltp1let 6147	Integer ordering relation.
|- ((M e. ZZ /\ N e. ZZ) -> (M < N <-> (M + 1) <_ N))
Theorem	zleltp1t 6148	Integer ordering relation.
|- ((M e. ZZ /\ N e. ZZ) -> (M <_ N <-> M < (N + 1)))
Theorem	zlem1ltt 6149	Integer ordering relation.
|- ((M e. ZZ /\ N e. ZZ) -> (M <_ N <-> (M - 1) < N))
Theorem	zltlem1t 6150	Integer ordering relation.
|- ((M e. ZZ /\ N e. ZZ) -> (M < N <-> M <_ (N - 1)))
Theorem	nn0lem1ltt 6151	Nonnegative integer ordering relation.
|- ((M e. NN0 /\ N e. NN0) -> (M <_ N <-> (M - 1) < N))
Theorem	nnlem1ltt 6152	Natural number ordering relation.
|- ((M e. NN /\ N e. NN) -> (M <_ N <-> (M - 1) < N))
Theorem	nnltlem1t 6153	Natural number ordering relation.
|- ((M e. NN /\ N e. NN) -> (M < N <-> M <_ (N - 1)))
Theorem	z2get 6154	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	zextlet 6155	An extensionality-like property for integer ordering.
|- ((M e. ZZ /\ N e. ZZ /\ A.k e. ZZ (k <_ M <-> k <_ N)) -> M = N)
Theorem	zextltt 6156	An extensionality-like property for integer ordering.
|- ((M e. ZZ /\ N e. ZZ /\ A.k e. ZZ (k < M <-> k < N)) -> M = N)
Theorem	recnzt 6157	The reciprocal of a number greater than 1 is not an integer.
|- ((A e. RR /\ 1 < A) -> -. (1 / A) e. ZZ)
Theorem	btwnnzt 6158	A number between an integer and its successor is not an integer.
|- ((A e. ZZ /\ A < B /\ B < (A + 1)) -> -. B e. ZZ)
Theorem	gtndivt 6159	A larger number does not divide a smaller natural number.
|- ((A e. RR /\ B e. NN /\ B < A) -> -. (B / A) e. ZZ)
Theorem	halfnz 6160	One-half is not an integer.
|- -. (1 / 2) e. ZZ
Theorem	primet 6161	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 6162	The square of a non-zero integer is a natural number.
|- ((A e. ZZ /\ A =/= 0) -> (A x. A) e. NN)
Theorem	nneo 6163	A natural number is even or odd but not both.
|- N e. NN   =>   |- ((N / 2) e. NN <-> -. ((N + 1) / 2) e. NN)
Theorem	nneot 6164	A natural number is even or odd but not both.
|- (N e. NN -> ((N / 2) e. NN <-> -. ((N + 1) / 2) e. NN))
Theorem	zeot 6165	An integer is even or odd.
|- (N e. ZZ -> ((N / 2) e. ZZ \/ ((N + 1) / 2) e. ZZ))
Theorem	zneo 6166	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	zneoOLD 6167	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 6168	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	dfuz 6169	An expression for the upper integers that start at N that is analogous to df-n 5892 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	peano5uz 6170	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	peano5uzt 6171	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 6172	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