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	peano2uz2 6201	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 6202	An expression for the upper integers that start at N that is analogous to df-n 5925 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 6203	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 6204	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 6205	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 6206	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 6207	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 6208	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 6209	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 6210	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 6211	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 6212	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 6213	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 6214	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 6215	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 6216	Lemma for uzwo3 6218 and zmin 6219.
Theorem	uzwo3lem2 6217	Lemma for uzwo3 6218.
Theorem	uzwo3 6218	Well-ordering principle: any non-empty subset of upper integers has a unique least element. This generalization of uzwo2 6457 allows the lower bound B to be any real number. See also nnwo 6458 and nnwos 6460.
|- ((B e. RR /\ (A (_ {z e. ZZ | B <_ z} /\ A =/= (/))) -> E!x e. A A.y e. A x <_ y)
Theorem	zmin 6219	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 6220	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 6221	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 6222	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 6223	Extend class notation with floor (greatest integer) function.
class |_
Definition	df-fl 6224	Define the floor (greatest integer) function. See flvalt 6225 for its value, flleltt 6228 for its basic property, and flclt 6226 for its closure. The term "floor" was coined by Ken Iverson. He also invented a mathematical notation for floor, consisting of an L-shaped left bracket and its reflection as a right bracket. In APL, the left-bracket alone is used, and we borrow this idea. (Thanks to Paul Chapman for this information.)
|- |_ = {<.x, y>. | (x e. RR /\ y = U.{z e. ZZ | (z <_ x /\ x < (z + 1))})}
Theorem	flvalt 6225	Value of the floor (greatest integer) function. The floor of A is the (unique) integer less than or equal to A whose successor is strictly greater than A.
|- (A e. RR -> (|_` A) = U.{x e. ZZ | (x <_ A /\ A < (x + 1))})
Theorem	flclt 6226	The floor (greatest integer) function is an integer (closure law).
|- (A e. RR -> (|_` A) e. ZZ)
Theorem	flreclt 6227	The floor (greatest integer) function is real.
|- (A e. RR -> (|_` A) e. RR)
Theorem	flleltt 6228	A basic property of the floor (greatest integer) function.
|- (A e. RR -> ((|_` A) <_ A /\ A < ((|_` A) + 1)))
Theorem	fllet 6229	A basic property of the floor (greatest integer) function.
|- (A e. RR -> (|_` A) <_ A)
Theorem	flltp1t 6230	A basic property of the floor (greatest integer) function.
|- (A e. RR -> A < ((|_` A) + 1))
Theorem	fraclt1t 6231	The fractional part of a real number is less than one.
|- (A e. RR -> (A - (|_` A)) < 1)
Theorem	fracge0t 6232	The fractional part of a real number is nonnegative.
|- (A e. RR -> 0 <_ (A - (|_` A)))
Theorem	flget 6233	The floor function value is the greatest integer less than or equal to its argument.
|- ((A e. RR /\ B e. ZZ) -> (B <_ A <-> B <_ (|_` A)))
Theorem	flltt 6234	The floor function value is less than the next integer.
|- ((A e. RR /\ B e. ZZ) -> (A < B <-> (|_` A) < B))
Theorem	flidt 6235	An integer is its own floor.
|- (A e. ZZ -> (|_` A) = A)
Theorem	flidmt 6236	The floor function is idempotent.
|- (A e. RR -> (|_` (|_` A)) = (|_` A))
Theorem	flwordit 6237	Ordering relationship for the greatest integer function.
|- ((A e. RR /\ B e. RR /\ A <_ B) -> (|_` A) <_ (|_` B))
Theorem	flval2t 6238	An alternate way to define the floor (greatest integer) function.
|- (A e. RR -> (|_` A) = U.{x e. ZZ | (x <_ A /\ A.y e. ZZ (y <_ A -> y <_ x))})
Theorem	flval3t 6239	An alternate way to define the floor (greatest integer) function, as the supremum of all integers less than or equal to its argument.
|- (A e. RR -> (|_` A) = sup({x e. ZZ | x <_ A}, RR, < ))
Theorem	flbit 6240	A condition equivalent to floor.
|- ((A e. RR /\ B e. ZZ) -> ((|_` A) = B <-> (B