mmtheorems65 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6401-6500 - Page 65 of 123

Type	Label	Description
Statement
Theorem	nn0ssq 6401	The nonnegative integers are a subset of the rationals.
|- NN0 (_ QQ
Theorem	nnssq 6402	The natural numbers are a subset of the rationals.
|- NN (_ QQ
Theorem	qssre 6403	The rationals are a subset of the reals.
|- QQ (_ RR
Theorem	qsscn 6404	The rationals are a subset of the complex numbers.
|- QQ (_ CC
Theorem	nnq 6405	A natural number is rational.
|- (A e. NN -> A e. QQ)
Theorem	qcn 6406	A rational number is a complex number.
|- (A e. QQ -> A e. CC)
Theorem	qex 6407	The set of rational numbers exists.
|- QQ e. V
Theorem	qaddcl 6408	Closure of addition of rationals.
|- ((A e. QQ /\ B e. QQ) -> (A + B) e. QQ)
Theorem	qnegcl 6409	Closure law for the negative of a rational.
|- (A e. QQ -> -uA e. QQ)
Theorem	qmulcl 6410	Closure of multiplication of rationals.
|- ((A e. QQ /\ B e. QQ) -> (A x. B) e. QQ)
Theorem	qsubcl 6411	Closure of subtraction of rationals.
|- ((A e. QQ /\ B e. QQ) -> (A - B) e. QQ)
Theorem	qreccl 6412	Closure of reciprocal of rationals.
|- ((A e. QQ /\ A =/= 0) -> (1 / A) e. QQ)
Theorem	qdivcl 6413	Closure of division of rationals.
|- ((A e. QQ /\ B e. QQ /\ B =/= 0) -> (A / B) e. QQ)
Theorem	qrevaddcl 6414	Reverse closure law for addition of rationals.
|- (B e. QQ -> ((A e. CC /\ (A + B) e. QQ) <-> A e. QQ))
Theorem	nnrecq 6415	The reciprocal of a natural number is rational.
|- (A e. NN -> (1 / A) e. QQ)
Theorem	irradd 6416	The sum of an irrational number and a rational number is irrational.
|- ((A e. (RR \ QQ) /\ B e. QQ) -> (A + B) e. (RR \ QQ))
Theorem	irrmul 6417	The product of an irrational with a nonzero rational is irrational.
|- ((A e. (RR \ QQ) /\ B e. QQ /\ B =/= 0) -> (A x. B) e. (RR \ QQ))
Theorem	qbtwnre 6418	The rational numbers are dense in RR: any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28.
|- ((A e. RR /\ B e. RR /\ A < B) -> E.x e. QQ (A < x /\ x < B))
Theorem	qbtwnxr 6419	The rational numbers are dense in RR*: any two extended real numbers have a rational between them.
|- ((A e. RR* /\ B e. RR* /\ A < B) -> E.x e. QQ (A < x /\ x < B))
Theorem	qsqueeze 6420	If a nonnegative real is less than any positive rational, it is zero.
|- ((A e. RR /\ 0 <_ A /\ A.x e. QQ (0 < x -> A < x)) -> A = 0)
The floor (greatest integer) function
Syntax	cfl 6421	Extend class notation with floor (greatest integer) function.
class |_
Definition	df-fl 6422	Define the floor (greatest integer) function. See flval 6423 for its value, fllelt 6426 for its basic property, and flcl 6424 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	flval 6423	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	flcl 6424	The floor (greatest integer) function is an integer (closure law).
|- (A e. RR -> (|_` A) e. ZZ)
Theorem	reflcl 6425	The floor (greatest integer) function is real.
|- (A e. RR -> (|_` A) e. RR)
Theorem	fllelt 6426	A basic property of the floor (greatest integer) function.
|- (A e. RR -> ((|_` A) <_ A /\ A < ((|_` A) + 1)))
Theorem	flle 6427	A basic property of the floor (greatest integer) function.
|- (A e. RR -> (|_` A) <_ A)
Theorem	flltp1 6428	A basic property of the floor (greatest integer) function.
|- (A e. RR -> A < ((|_` A) + 1))
Theorem	fraclt1 6429	The fractional part of a real number is less than one.
|- (A e. RR -> (A - (|_` A)) < 1)
Theorem	fracge0 6430	The fractional part of a real number is nonnegative.
|- (A e. RR -> 0 <_ (A - (|_` A)))
Theorem	flge 6431	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	fllt 6432	The floor function value is less than the next integer.
|- ((A e. RR /\ B e. ZZ) -> (A < B <-> (|_` A) < B))
Theorem	flid 6433	An integer is its own floor.
|- (A e. ZZ -> (|_` A) = A)
Theorem	flidm 6434	The floor function is idempotent.
|- (A e. RR -> (|_` (|_` A)) = (|_` A))
Theorem	flidz 6435	A real number equals its floor iff it is an integer.
|- (A e. RR -> ((|_` A) = A <-> A e. ZZ))
Theorem	flwordi 6436	Ordering relationship for the greatest integer function.
|- ((A e. RR /\ B e. RR /\ A <_ B) -> (|_` A) <_ (|_` B))
Theorem	flval2 6437	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	flval3 6438	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	flbi 6439	A condition equivalent to floor.
|- ((A e. RR /\ B e. ZZ) -> ((|_` A) = B <-> (B <_ A /\ A < (B + 1))))
Theorem	flbi2 6440	A condition equivalent to floor.
|- ((N e. ZZ /\ F e. RR) -> ((|_` (N + F)) = N <-> (0 <_ F /\ F < 1)))
Theorem	flge0nn0 6441	The floor of a number greater than or equal to 0 is a nonnegative integer.
|- ((A e. RR /\ 0 <_ A) -> (|_` A) e. NN0)
Theorem	flge1nn 6442	The floor of a number greater than or equal to 1 is a natural number.
|- ((A e. RR /\ 1 <_ A) -> (|_` A) e. NN)
Theorem	fladdz 6443	An integer can be moved in and out of the floor of a sum.
|- ((A e. RR /\ N e. ZZ) -> (|_` (A + N)) = ((|_` A) + N))
Theorem	flzadd 6444	An integer can be moved in and out of the floor of a sum.
|- ((N e. ZZ /\ A e. RR) -> (|_` (N + A)) = (N + (|_` A)))
Theorem	btwnzge0 6445	A real bounded between an integer and its successor is nonnegative iff the integer is nonnegative. Second half of Lemma 13-4.1 of [Gleason] p. 217. (For the first half see rebtwnz 6394.)
|- (((A e. RR /\ N e. ZZ) /\ (N <_ A /\ A < (N + 1))) -> (0 <_ A <-> 0 <_ N))
Theorem	flhalf 6446	Ordering relation for the floor of half of an integer.
|- (N e. ZZ -> N <_ (2 x. (|_` ((N + 1) / 2))))
Theorem	ceicl 6447	The ceiling function returns an integer (closure law). (Contributed by Jeffrey Hankins, 10-Jun-2007.)
|- (A e. RR -> -u(|_` -uA) e. ZZ)
Theorem	ceige 6448	The ceiling of a real number is greater than or equal to that number. (Contributed by Jeffrey Hankins, 10-Jun-2007.)
|- (A e. RR -> A <_ -u(|_` -uA))
Theorem	ceim1l 6449	One less than the ceiling of a real number is strictly less than that number. (Contributed by Jeffrey Hankins, 10-Jun-2007.)
|- (A e. RR -> (-u(|_` -uA) - 1) < A)
Theorem	ceile 6450	The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jeffrey Hankins, 10-Jun-2007.)
|- ((A e. RR /\ B e. ZZ /\ A <_ B) -> -u(|_` -uA) <_ B)
Theorem	quoremz 6451	Quotient and remainder of an integer divided by a natural number.
|- Q = (|_` (A / B))   &   |- R = (A - (B x. Q))   =>   |- ((A e. ZZ /\ B e. NN) -> ((Q e. ZZ /\ R e. NN0) /\ (R < B /\ A = ((B x. Q) + R))))
Theorem	quoremnn0ALT 6452	Quotient and remainder of a nonnegative integer divided by a natural number.
|- Q = (|_` (A / B))   &   |- R = (A - (B x. Q))   =>   |- ((A e. NN0 /\ B e. NN) -> ((Q e. NN0 /\ R e. NN0) /\ (R < B /\ A = ((B x. Q) + R))))
Theorem	quoremnn0 6453	Quotient and remainder of a nonnegative integer divided by a natural number.
|- Q = (|_` (A / B))   &   |- R = (A - (B x. Q))   =>   |- ((A e. NN0 /\ B e. NN) -> ((Q e. NN0 /\ R e. NN0) /\ (R < B /\ A = ((B x. Q) + R))))
Theorem	intfrac2 6454	Decompose a real into integer and fractional parts.
|- Z = (|_` A)   &   |- F = (A - Z)   =>   |- (A e. RR -> (0 <_ F /\ F < 1 /\ A = (Z + F)))
Theorem	intfracq 6455	Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 6454.
|- Z = (|_` (M / N))   &   |- F = ((M / N) - Z)   =>   |- ((M e. ZZ /\ N e. NN) -> (0 <_ F /\ F <_ ((N - 1) / N) /\ (M / N) = (Z + F)))
Theorem	fldiv 6456	Cancellation of the embedded floor of a real divided by an integer.
|- ((A e. RR /\ N e. NN) -> (|_` ((|_` A) / N)) = (|_` (A / N)))
Theorem	fldiv2 6457	Cancellation of an embedded floor of a ratio. Generalization of Equation 2.4 in [CormenLeisersonRivest] p. 33 (where A must be an integer).
|- ((A e. RR /\ M e. NN /\ N e. NN) -> (|_` ((|_` (A / M)) / N)) = (|_` (A / (M x. N))))
The modulo (remainder) operation
Syntax	cmo 6458	Extend class notation with the modulo operation.
class mod
Definition	df-mod 6459	Define the modulo (remainder) operation. See modval 6460 for its value.
|- mod = {<.<.x, y>., z>. | ((x e. RR /\ y e. RR+) /\ z = (x - (y x. (|_` (x / y)))))}
Theorem	modval 6460	The value of the modulo operation. The modulo congruence notation of number theory, J == K( modulo N), can be expressed in our notation as (J mod N) = (K mod N). Definition 1 in Knuth, The Art of Computer Programming, Vol. I (1972), p. 38. Knuth uses "mod" for the operation and "modulo" for the congruence. Unlike Knuth, we restrict the second argument to positive reals to simplify certain theorems. (This also gives us future flexibility to extend it to any one of several different conventions for a zero or negative second argument, should there be an advantage in doing so.)
|- ((A e. RR /\ B e. RR+) -> (A mod B) = (A - (B x. (|_` (A / B)))))
Theorem	modcl 6461	Closure law for the modulo operation.
|- ((A e. RR /\ B e. RR+) -> (A mod B) e. RR)
Theorem	modge0 6462	The modulo operation is nonnegative.
|- ((A e. RR /\ B e. RR+) -> 0 <_ (A mod B))
Theorem	modlt 6463	The modulo operation is less than its second argument.
|- ((A e. RR /\ B e. RR+) -> (A mod B) < B)
Theorem	modfrac 6464	The fractional part of a number is the number modulo 1.
|- (A e. RR -> (A mod 1) = (A - (|_` A)))
Theorem	flmod 6465	The floor function expressed in terms of the modulo operation.
|- (A e. RR -> (|_` A) = (A - (A mod 1)))
Theorem	intfrac 6466	Break a number into its integer part and its fractional part.
|- (A e. RR -> A = ((|_` A) + (A mod 1)))
Theorem	flmulnn0 6467	Move a nonnegative integer in and out of a floor.
|- ((N e. NN0 /\ A e. RR) -> (N x. (|_` A)) <_ (|_` (N x. A)))
Theorem	flmulnn0OLD 6468	Move a nonnegative integer in and out of a floor.
|- ((N e. NN0 /\ A e. RR) -> (N x. (|_` A)) <_ (|_` (N x. A)))
Theorem	modmulnn 6469	Move a natural number in and out of a floor in the first argument of a modulo operation.
|- ((N e. NN /\ A e. RR /\ M e. NN) -> ((N x. (|_` A)) mod (N x. M)) <_ ((|_` (N x. A)) mod (N x. M)))
Theorem	zmodcl 6470	Closure law for the modulo operation restricted to integers.
|- ((A e. ZZ /\ B e. NN) -> (A mod B) e. NN0)
Theorem	modid 6471	Identity law for modulo.
|- (((A e. RR /\ B e. RR+) /\ (0 <_ A /\ A < B)) -> (A mod B) = A)
Theorem	modid2 6472	Identity law for modulo.
|- ((A e. RR /\ B e. RR+) -> ((A mod B) = A <-> (0 <_ A /\ A < B)))
Theorem	modabs 6473	Absorption law for modulo.
|- (((A e. RR /\ B e. RR+ /\ C e. RR+) /\ B <_ C) -> ((A mod B) mod C) = (A mod B))
Theorem	modabs2 6474	Absorption law for modulo.
|- ((A e. RR /\ B e. RR+) -> ((A mod B) mod B) = (A mod B))
Theorem	modcyc 6475	The modulo operation is periodic.
|- ((A e. RR /\ B e. RR+ /\ N e. ZZ) -> ((A + (N x. B)) mod B) = (A mod B))
Theorem	modcyc2 6476	The modulo operation is periodic.
|- ((A e. RR /\ B e. RR+ /\ N e. ZZ) -> ((A - (B x. N)) mod B) = (A mod B))
Theorem	modadd1 6477	Addition property of the modulo operation.
|- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR+) /\ (A mod D) = (B mod D)) -> ((A + C) mod D) = ((B + C) mod D))
Theorem	modmul1 6478	Multiplication property of the modulo operation. Note that the multiplier C must be an integer.
|- (((A e. RR /\ B e. RR) /\ (C e. ZZ /\ D e. RR+) /\ (A mod D) = (B mod D)) -> ((A x. C) mod D) = ((B x. C) mod D))
Theorem	moddi 6479	Distribute multiplication over a modulo operation.
|- ((A e. RR+ /\ B e. RR /\ C e. RR+) -> (A x. (B mod C)) = ((A x. B) mod (A x. C)))
Theorem	modsubdir 6480	Distribute the modulo operation over a subtraction.
|- ((A e. RR /\ B e. RR /\ C e. RR+) -> ((B mod C) <_ (A mod C) <-> ((A - B) mod C) = ((A mod C) - (B mod C))))
Theorem	modirr 6481	A number modulo an irrational multiple of it is nonzero.
|- ((A e. RR /\ B e. RR+ /\ (A / B) e. (RR \ QQ)) -> (A mod B) =/= 0)
Monotonic sequences
Theorem	monoord 6482	Ordering relation for a monotonic sequence.
|- F:NN-->RR   &   |- (x e. NN -> (F` x) <_ (F` (x + 1)))   =>   |- ((A e. NN /\ B e. NN /\ A <_ B) -> (F` A) <_ (F` B))
Real number intervals
Syntax	cioo 6483	Extend class notation with the set of open intervals of extended reals.
class (,)
Syntax	cioc 6484	Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
Syntax	cico 6485	Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
Syntax	cicc 6486	Extend class notation with the set of closed intervals of extended reals.
class [,]
Definition	df-ioo 6487	Define the set of open intervals of extended reals.
|- (,) = {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})}
Definition	df-ioc 6488	Define the set of open-below, closed-above intervals of extended reals.
|- (,] = {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w <_ y)})}
Definition	df-ico 6489	Define the set of closed-below, open-above intervals of extended reals.
|- [,) = {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x <_ w /\ w < y)})}
Definition	df-icc 6490	Define the set of closed intervals of extended reals.
|- [,] = {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x <_ w /\ w <_ y)})}
Theorem	iooex 6491	The set of open intervals of extended reals exists.
|- (,) e. V
Theorem	iooval 6492	Value of the open interval function.
|- ((A e. RR* /\ B e. RR*) -> (A(,)B) = {x e. RR* | (A < x /\ x < B)})
Theorem	iooval2 6493	Value of the open interval function.
|- ((A e. RR* /\ B e. RR*) -> (A(,)B) = {x e. RR | (A < x /\ x < B)})
Theorem	ioo0 6494	An empty open interval of extended reals.
|- ((A e. RR* /\ B e. RR*) -> ((A(,)B) = (/) <-> B <_ A))
Theorem	ioon0 6495	An open interval of extended reals is nonempty iff the lower argument is less than the upper argument.
|- ((A e. RR* /\ B e. RR*) -> ((A(,)B) =/= (/) <-> A < B))
Theorem	ndmioo 6496	The open interval function's value is empty outside of its domain.
|- (-. (A e. RR* /\ B e. RR*) -> (A(,)B) = (/))
Theorem	iooid 6497	An open interval with identical lower and upper bounds is empty.
|- (A(,)A) = (/)
Theorem	iooin 6498	Intersection of two open intervals of extended reals.
|- (((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ D e. RR*)) -> ((A(,)B) i^i (C(,)D)) = (if(A <_ C, C, A)(,)if(B <_ D, B, D)))
Theorem	iooss1 6499	Subset relationship for open intervals of extended reals.
|- (((A e. RR* /\ B e. RR* /\ C e. RR*) /\ A <_ B) -> (B(,)C) (_ (A(,)C))
Theorem	iooss2 6500	Subset relationship for open intervals of extended reals.
|- (((A e. RR* /\ B e. RR* /\ C e. RR*) /\ B <_ C) -> (A(,)B) (_ (A(,)C))