P. 100 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9901-10000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elfzomelpfzo 9901	An integer increased by another integer is an element of a half-open integer range if and only if the integer is contained in the half-open integer range with bounds decreased by the other integer. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ L e. ZZ ) ) -> ( K e. ( ( M - L )..^ ( N - L ) ) <-> ( K + L ) e. ( M..^ N ) ) )
Theorem	peano2fzor 9902	A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.)
|- ( ( K e. ( ZZ>= ` M ) /\ ( K + 1 ) e. ( M..^ N ) ) -> K e. ( M..^ N ) )
Theorem	fzosplitsn 9903	Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
|- ( B e. ( ZZ>= ` A ) -> ( A..^ ( B + 1 ) ) = ( ( A..^ B ) u. { B } ) )
Theorem	fzosplitprm1 9904	Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
|- ( ( A e. ZZ /\ B e. ZZ /\ A < B ) -> ( A..^ ( B + 1 ) ) = ( ( A..^ ( B - 1 ) ) u. { ( B - 1 ) , B } ) )
Theorem	fzosplitsni 9905	Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
|- ( B e. ( ZZ>= ` A ) -> ( C e. ( A..^ ( B + 1 ) ) <-> ( C e. ( A..^ B ) \/ C = B ) ) )
Theorem	fzisfzounsn 9906	A finite interval of integers as union of a half-open integer range and a singleton. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
|- ( B e. ( ZZ>= ` A ) -> ( A ... B ) = ( ( A..^ B ) u. { B } ) )
Theorem	fzostep1 9907	Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
|- ( A e. ( B..^ C ) -> ( ( A + 1 ) e. ( B..^ C ) \/ ( A + 1 ) = C ) )
Theorem	fzoshftral 9908*	Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 9781. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
|- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( A. j e. ( M..^ N ) ph <-> A. k e. ( ( M + K )..^ ( N + K ) ) [. ( k - K ) / j ]. ph ) )
Theorem	fzind2 9909*	Induction on the integers from M to N inclusive. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Version of fzind 9070 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.)
|- ( x = M -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> th ) )   &    |- ( x = K -> ( ph <-> ta ) )   &    |- ( N e. ( ZZ>= ` M ) -> ps )   &    |- ( y e. ( M..^ N ) -> ( ch -> th ) )   =>    |- ( K e. ( M ... N ) -> ta )
Theorem	exfzdc 9910*	Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ n e. ( M ... N ) ) -> DECID ps )   =>    |- ( ph -> DECID E. n e. ( M ... N ) ps )
Theorem	fvinim0ffz 9911	The function values for the borders of a finite interval of integers, which is the domain of the function, are not in the image of the interior of the interval iff the intersection of the images of the interior and the borders is empty. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 5-Feb-2021.)
|- ( ( F : ( 0 ... K ) --> V /\ K e. NN0 ) -> ( ( ( F " { 0 , K } ) i^i ( F " ( 1..^ K ) ) ) = (/) <-> ( ( F ` 0 ) e/ ( F " ( 1..^ K ) ) /\ ( F ` K ) e/ ( F " ( 1..^ K ) ) ) ) )
Theorem	subfzo0 9912	The difference between two elements in a half-open range of nonnegative integers is greater than the negation of the upper bound and less than the upper bound of the range. (Contributed by AV, 20-Mar-2021.)
|- ( ( I e. ( 0..^ N ) /\ J e. ( 0..^ N ) ) -> ( -u N < ( I - J ) /\ ( I - J ) < N ) )
4.5.7  Rational numbers (cont.)
Theorem	qtri3or 9913	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
|- ( ( M e. QQ /\ N e. QQ ) -> ( M < N \/ M = N \/ N < M ) )
Theorem	qletric 9914	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A <_ B \/ B <_ A ) )
Theorem	qlelttric 9915	Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A <_ B \/ B < A ) )
Theorem	qltnle 9916	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A < B <-> -. B <_ A ) )
Theorem	qdceq 9917	Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A = B )
Theorem	exbtwnzlemstep 9918*	Lemma for exbtwnzlemex 9920. Induction step. (Contributed by Jim Kingdon, 10-May-2022.)
|- ( ph -> K e. NN )   &    |- ( ph -> A e. RR )   &    |- ( ( ph /\ n e. ZZ ) -> ( n <_ A \/ A < n ) )   =>    |- ( ( ph /\ E. m e. ZZ ( m <_ A /\ A < ( m + ( K + 1 ) ) ) ) -> E. m e. ZZ ( m <_ A /\ A < ( m + K ) ) )
Theorem	exbtwnzlemshrink 9919*	Lemma for exbtwnzlemex 9920. Shrinking the range around A. (Contributed by Jim Kingdon, 10-May-2022.)
|- ( ph -> J e. NN )   &    |- ( ph -> A e. RR )   &    |- ( ( ph /\ n e. ZZ ) -> ( n <_ A \/ A < n ) )   =>    |- ( ( ph /\ E. m e. ZZ ( m <_ A /\ A < ( m + J ) ) ) -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
Theorem	exbtwnzlemex 9920*	Existence of an integer so that a given real number is between the integer and its successor. The real number must satisfy the n <_ A \/ A < n hypothesis. For example either a rational number or a number which is irrational (in the sense of being apart from any rational number) will meet this condition. The proof starts by finding two integers which are less than and greater than A. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on the n <_ A \/ A < n hypothesis, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ph -> A e. RR )   &    |- ( ( ph /\ n e. ZZ ) -> ( n <_ A \/ A < n ) )   =>    |- ( ph -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
Theorem	exbtwnz 9921*	If a real number is between an integer and its successor, there is a unique greatest integer less than or equal to the real number. (Contributed by Jim Kingdon, 10-May-2022.)
|- ( ph -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )   &    |- ( ph -> A e. RR )   =>    |- ( ph -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
Theorem	qbtwnz 9922*	There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
Theorem	rebtwn2zlemstep 9923*	Lemma for rebtwn2z 9925. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.)
|- ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR /\ E. m e. ZZ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> E. m e. ZZ ( m < A /\ A < ( m + K ) ) )
Theorem	rebtwn2zlemshrink 9924*	Lemma for rebtwn2z 9925. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.)
|- ( ( A e. RR /\ J e. ( ZZ>= ` 2 ) /\ E. m e. ZZ ( m < A /\ A < ( m + J ) ) ) -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )
Theorem	rebtwn2z 9925*	A real number can be bounded by integers above and below which are two apart. The proof starts by finding two integers which are less than and greater than the given real number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.)
|- ( A e. RR -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )
Theorem	qbtwnrelemcalc 9926	Lemma for qbtwnre 9927. Calculations involved in showing the constructed rational number is less than B. (Contributed by Jim Kingdon, 14-Oct-2021.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. NN )   &    |- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> M < ( A x. ( 2 x. N ) ) )   &    |- ( ph -> ( 1 / N ) < ( B - A ) )   =>    |- ( ph -> ( ( M + 2 ) / ( 2 x. N ) ) < B )
Theorem	qbtwnre 9927*	The rational numbers are dense in RR: any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.)
|- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
Theorem	qbtwnxr 9928*	The rational numbers are dense in RR*: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
Theorem	qavgle 9929	The average of two rational numbers is less than or equal to at least one of them. (Contributed by Jim Kingdon, 3-Nov-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( ( ( A + B ) / 2 ) <_ A \/ ( ( A + B ) / 2 ) <_ B ) )
Theorem	ioo0 9930	An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) = (/) <-> B <_ A ) )
Theorem	ioom 9931*	An open interval of extended reals is inhabited iff the lower argument is less than the upper argument. (Contributed by Jim Kingdon, 27-Nov-2021.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( E. x x e. ( A (,) B ) <-> A < B ) )
Theorem	ico0 9932	An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <-> B <_ A ) )
Theorem	ioc0 9933	An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,] B ) = (/) <-> B <_ A ) )
4.6  Elementary integer functions
4.6.1  The floor and ceiling functions
Syntax	cfl 9934	Extend class notation with floor (greatest integer) function.
class |_
Syntax	cceil 9935	Extend class notation to include the ceiling function.
class ⌈
Definition	df-fl 9936*	Define the floor (greatest integer less than or equal to) function. See flval 9938 for its value, flqlelt 9942 for its basic property, and flqcl 9939 for its closure. For example, ( |_ ` ( 3 / 2 ) ) = 1 while ( |_ ` -u ( 3 / 2 ) ) = -u 2 (ex-fl 12630). Although we define this on real numbers so that notations are similar to the Metamath Proof Explorer, in the absence of excluded middle few theorems will be possible for all real numbers. Imagine a real number which is around 2.99995 or 3.00001 . In order to determine whether its floor is 2 or 3, it would be necessary to compute the number to arbitrary precision. 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.) (Contributed by NM, 14-Nov-2004.)
|- |_ = ( x e. RR |-> ( iota_ y e. ZZ ( y <_ x /\ x < ( y + 1 ) ) ) )
Definition	df-ceil 9937	The ceiling (least integer greater than or equal to) function. Defined in ISO 80000-2:2009(E) operation 2-9.18 and the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. See ceilqval 9972 for its value, ceilqge 9976 and ceilqm1lt 9978 for its basic properties, and ceilqcl 9974 for its closure. For example, (⌈ ` ( 3 / 2 ) ) = 2 while (⌈ ` -u ( 3 / 2 ) ) = -u 1 (ex-ceil 12631). As described in df-fl 9936 most theorems are only for rationals, not reals. The symbol ⌈ is inspired by the gamma shaped left bracket of the usual notation. (Contributed by David A. Wheeler, 19-May-2015.)
|- ⌈ = ( x e. RR |-> -u ( |_ ` -u x ) )
Theorem	flval 9938*	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. (Contributed by NM, 14-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
|- ( A e. RR -> ( |_ ` A ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) )
Theorem	flqcl 9939	The floor (greatest integer) function yields an integer when applied to a rational (closure law). For a similar closure law for real numbers apart from any integer, see flapcl 9941. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) e. ZZ )
Theorem	apbtwnz 9940*	There is a unique greatest integer less than or equal to a real number which is apart from all integers. (Contributed by Jim Kingdon, 11-May-2022.)
|- ( ( A e. RR /\ A. n e. ZZ A # n ) -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
Theorem	flapcl 9941*	The floor (greatest integer) function yields an integer when applied to a real number apart from any integer. For example, an irrational number (see for example sqrt2irrap 11703) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.)
|- ( ( A e. RR /\ A. n e. ZZ A # n ) -> ( |_ ` A ) e. ZZ )
Theorem	flqlelt 9942	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) )
Theorem	flqcld 9943	The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ph -> A e. QQ )   =>    |- ( ph -> ( |_ ` A ) e. ZZ )
Theorem	flqle 9944	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) <_ A )
Theorem	flqltp1 9945	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> A < ( ( |_ ` A ) + 1 ) )
Theorem	qfraclt1 9946	The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( A - ( |_ ` A ) ) < 1 )
Theorem	qfracge0 9947	The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> 0 <_ ( A - ( |_ ` A ) ) )
Theorem	flqge 9948	The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ( A e. QQ /\ B e. ZZ ) -> ( B <_ A <-> B <_ ( |_ ` A ) ) )
Theorem	flqlt 9949	The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ( A e. QQ /\ B e. ZZ ) -> ( A < B <-> ( |_ ` A ) < B ) )
Theorem	flid 9950	An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
|- ( A e. ZZ -> ( |_ ` A ) = A )
Theorem	flqidm 9951	The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` ( |_ ` A ) ) = ( |_ ` A ) )
Theorem	flqidz 9952	A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.)
|- ( A e. QQ -> ( ( |_ ` A ) = A <-> A e. ZZ ) )
Theorem	flqltnz 9953	If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.)
|- ( ( A e. QQ /\ -. A e. ZZ ) -> ( |_ ` A ) < A )
Theorem	flqwordi 9954	Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ A <_ B ) -> ( |_ ` A ) <_ ( |_ ` B ) )
Theorem	flqword2 9955	Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ A <_ B ) -> ( |_ ` B ) e. ( ZZ>= ` ( |_ ` A ) ) )
Theorem	flqbi 9956	A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
|- ( ( A e. QQ /\ B e. ZZ ) -> ( ( |_ ` A ) = B <-> ( B <_ A /\ A < ( B + 1 ) ) ) )
Theorem	flqbi2 9957	A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
|- ( ( N e. ZZ /\ F e. QQ ) -> ( ( |_ ` ( N + F ) ) = N <-> ( 0 <_ F /\ F < 1 ) ) )
Theorem	adddivflid 9958	The floor of a sum of an integer and a fraction is equal to the integer iff the denominator of the fraction is less than the numerator. (Contributed by AV, 14-Jul-2021.)
|- ( ( A e. ZZ /\ B e. NN0 /\ C e. NN ) -> ( B < C <-> ( |_ ` ( A + ( B / C ) ) ) = A ) )
Theorem	flqge0nn0 9959	The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( ( A e. QQ /\ 0 <_ A ) -> ( |_ ` A ) e. NN0 )
Theorem	flqge1nn 9960	The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( ( A e. QQ /\ 1 <_ A ) -> ( |_ ` A ) e. NN )
Theorem	fldivnn0 9961	The floor function of a division of a nonnegative integer by a positive integer is a nonnegative integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
|- ( ( K e. NN0 /\ L e. NN ) -> ( |_ ` ( K / L ) ) e. NN0 )
Theorem	divfl0 9962	The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.)
|- ( ( A e. NN0 /\ B e. NN ) -> ( A < B <-> ( |_ ` ( A / B ) ) = 0 ) )
Theorem	flqaddz 9963	An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( ( A e. QQ /\ N e. ZZ ) -> ( |_ ` ( A + N ) ) = ( ( |_ ` A ) + N ) )
Theorem	flqzadd 9964	An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( ( N e. ZZ /\ A e. QQ ) -> ( |_ ` ( N + A ) ) = ( N + ( |_ ` A ) ) )
Theorem	flqmulnn0 9965	Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( ( N e. NN0 /\ A e. QQ ) -> ( N x. ( |_ ` A ) ) <_ ( |_ ` ( N x. A ) ) )
Theorem	btwnzge0 9966	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. (Contributed by NM, 12-Mar-2005.)
|- ( ( ( A e. RR /\ N e. ZZ ) /\ ( N <_ A /\ A < ( N + 1 ) ) ) -> ( 0 <_ A <-> 0 <_ N ) )
Theorem	2tnp1ge0ge0 9967	Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.)
|- ( N e. ZZ -> ( 0 <_ ( ( 2 x. N ) + 1 ) <-> 0 <_ N ) )
Theorem	flhalf 9968	Ordering relation for the floor of half of an integer. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
|- ( N e. ZZ -> N <_ ( 2 x. ( |_ ` ( ( N + 1 ) / 2 ) ) ) )
Theorem	fldivnn0le 9969	The floor function of a division of a nonnegative integer by a positive integer is less than or equal to the division. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
|- ( ( K e. NN0 /\ L e. NN ) -> ( |_ ` ( K / L ) ) <_ ( K / L ) )
Theorem	flltdivnn0lt 9970	The floor function of a division of a nonnegative integer by a positive integer is less than the division of a greater dividend by the same positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
|- ( ( K e. NN0 /\ N e. NN0 /\ L e. NN ) -> ( K < N -> ( |_ ` ( K / L ) ) < ( N / L ) ) )
Theorem	fldiv4p1lem1div2 9971	The floor of an integer equal to 3 or greater than 4, increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
|- ( ( N = 3 \/ N e. ( ZZ>= ` 5 ) ) -> ( ( |_ ` ( N / 4 ) ) + 1 ) <_ ( ( N - 1 ) / 2 ) )
Theorem	ceilqval 9972	The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( A e. QQ -> (⌈ ` A ) = -u ( |_ ` -u A ) )
Theorem	ceiqcl 9973	The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> -u ( |_ ` -u A ) e. ZZ )
Theorem	ceilqcl 9974	Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> (⌈ ` A ) e. ZZ )
Theorem	ceiqge 9975	The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> A <_ -u ( |_ ` -u A ) )
Theorem	ceilqge 9976	The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> A <_ (⌈ ` A ) )
Theorem	ceiqm1l 9977	One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> ( -u ( |_ ` -u A ) - 1 ) < A )
Theorem	ceilqm1lt 9978	One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> ( (⌈ ` A ) - 1 ) < A )
Theorem	ceiqle 9979	The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( ( A e. QQ /\ B e. ZZ /\ A <_ B ) -> -u ( |_ ` -u A ) <_ B )
Theorem	ceilqle 9980	The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( ( A e. QQ /\ B e. ZZ /\ A <_ B ) -> (⌈ ` A ) <_ B )
Theorem	ceilid 9981	An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
|- ( A e. ZZ -> (⌈ ` A ) = A )
Theorem	ceilqidz 9982	A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> ( A e. ZZ <-> (⌈ ` A ) = A ) )
Theorem	flqleceil 9983	The floor of a rational number is less than or equal to its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) <_ (⌈ ` A ) )
Theorem	flqeqceilz 9984	A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> ( A e. ZZ <-> ( |_ ` A ) = (⌈ ` A ) ) )
Theorem	intqfrac2 9985	Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- Z = ( |_ ` A )   &    |- F = ( A - Z )   =>    |- ( A e. QQ -> ( 0 <_ F /\ F < 1 /\ A = ( Z + F ) ) )
Theorem	intfracq 9986	Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 9985. (Contributed by NM, 16-Aug-2008.)
|- Z = ( |_ ` ( M / N ) )   &    |- F = ( ( M / N ) - Z )   =>    |- ( ( M e. ZZ /\ N e. NN ) -> ( 0 <_ F /\ F <_ ( ( N - 1 ) / N ) /\ ( M / N ) = ( Z + F ) ) )
Theorem	flqdiv 9987	Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ N e. NN ) -> ( |_ ` ( ( |_ ` A ) / N ) ) = ( |_ ` ( A / N ) ) )
4.6.2  The modulo (remainder) operation
Syntax	cmo 9988	Extend class notation with the modulo operation.
class mod
Definition	df-mod 9989*	Define the modulo (remainder) operation. See modqval 9990 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 9936 we define this for first and second arguments which are real and positive real, respectively, even though many theorems will need to be more restricted (for example, specify rational arguments). (Contributed by NM, 10-Nov-2008.)
|- mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
Theorem	modqval 9990	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 numbers 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.) As with flqcl 9939 we only prove this for rationals although other particular kinds of real numbers may be possible. (Contributed by Jim Kingdon, 16-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
Theorem	modqvalr 9991	The value of the modulo operation (multiplication in reversed order). (Contributed by Jim Kingdon, 16-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( ( |_ ` ( A / B ) ) x. B ) ) )
Theorem	modqcl 9992	Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) e. QQ )
Theorem	flqpmodeq 9993	Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( ( |_ ` ( A / B ) ) x. B ) + ( A mod B ) ) = A )
Theorem	modqcld 9994	Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> 0 < B )   =>    |- ( ph -> ( A mod B ) e. QQ )
Theorem	modq0 9995	A mod B is zero iff A is evenly divisible by B. (Contributed by Jim Kingdon, 17-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( A / B ) e. ZZ ) )
Theorem	mulqmod0 9996	The product of an integer and a positive rational number is 0 modulo the positive real number. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) -> ( ( A x. M ) mod M ) = 0 )
Theorem	negqmod0 9997	A is divisible by B iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( -u A mod B ) = 0 ) )
Theorem	modqge0 9998	The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 <_ ( A mod B ) )
Theorem	modqlt 9999	The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) < B )
Theorem	modqelico 10000	Modular reduction produces a half-open interval. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) e. ( 0 [,) B ) )