mmtheorems63 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6201-6300 - Page 63 of 108

Type	Label	Description
Statement
Theorem	nnltlem1t 6201	Natural number ordering relation.
|- ((M e. NN /\ N e. NN) -> (M < N <-> M <_ (N - 1)))
Theorem	zdivt 6202	Two ways to express "A divides B.
|- ((A e. NN /\ B e. ZZ) -> (E.x e. ZZ (A x. x) = B <-> (B / A) e. ZZ))
Theorem	z2get 6203	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 6204	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 6205	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 6206	The reciprocal of a number greater than 1 is not an integer.
|- ((A e. RR /\ 1 < A) -> -. (1 / A) e. ZZ)
Theorem	btwnnzt 6207	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 6208	A larger number does not divide a smaller natural number.
|- ((A e. RR /\ B e. NN /\ B < A) -> -. (B / A) e. ZZ)
Theorem	halfnz 6209	One-half is not an integer.
|- -. (1 / 2) e. ZZ
Theorem	primet 6210	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 6211	The square of a non-zero integer is a natural number.
|- ((A e. ZZ /\ A =/= 0) -> (A x. A) e. NN)
Theorem	nneo 6212	A natural number is even or odd but not both.
|- N e. NN   =>   |- ((N / 2) e. NN <-> -. ((N + 1) / 2) e. NN)
Theorem	nneot 6213	A natural number is even or odd but not both.
|- (N e. NN -> ((N / 2) e. NN <-> -. ((N + 1) / 2) e. NN))
Theorem	zeot 6214	An integer is even or odd.
|- (N e. ZZ -> ((N / 2) e. ZZ \/ ((N + 1) / 2) e. ZZ))
Theorem	zneo 6215	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 6216	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 6217	An expression for the upper integers that start at N that is analogous to df-n 5939 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 6218	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 6219	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 6220	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 6221	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 6222	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 6223	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 6224	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 6225	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 6226	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 6227	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 6228	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 6229	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	btwnz 6230	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 6231	Lemma for uzwo3 6233 and zmin 6234.
Theorem	uzwo3lem2 6232	Lemma for uzwo3 6233.
Theorem	uzwo3 6233	Well-ordering principle: any non-empty subset of upper integers has a unique least element. This generalization of uzwo2 6471 allows the lower bound B to be any real number. See also nnwo 6472 and nnwos 6474.
|- ((B e. RR /\ (A (_ {z e. ZZ | B <_ z} /\ A =/= (/))) -> E!x e. A A.y e. A x <_ y)
Theorem	zmin 6234	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 6235	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 6236	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 6237	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)))
The floor (greatest integer) function
Syntax	cfl 6238	Extend class notation with floor (greatest integer) function.
class |_
Definition	df-fl 6239	Define the floor (greatest integer) function. See flvalt 6240 for its value, flleltt 6243 for its basic property, and flclt 6241 for its closure. The term "floor" was coined by Ken Iverson. He also invented a mathematical notation for floor, consisting of