P. 108 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10701-10800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	exbtwnzlemex 10701*	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 10702*	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 10703*	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 10704*	Lemma for rebtwn2z 10706. 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 10705*	Lemma for rebtwn2z 10706. 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 10706*	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 10707	Lemma for qbtwnre 10708. 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 10708*	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 10709*	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 10710	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 10711	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 10712*	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 10713	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 10714	An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,] B ) = (/) <-> B <_ A ) )
Theorem	dfrp2 10715	Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
|- RR+ = ( 0 (,) +oo )
Theorem	elicod 10716	Membership in a left-closed right-open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> C e. RR* )   &    |- ( ph -> A <_ C )   &    |- ( ph -> C < B )   =>    |- ( ph -> C e. ( A [,) B ) )
Theorem	icogelb 10717	An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,) B ) ) -> A <_ C )
Theorem	elicore 10718	A member of a left-closed right-open interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C e. RR )
Theorem	xqltnle 10719	"Less than" expressed in terms of "less than or equal to", for extended numbers which are rational or +oo. We have not yet had enough usage of such numbers to warrant fully developing the concept, as in NN0* or RR*, so for now we just have a handful of theorems for what we need. (Contributed by Jim Kingdon, 5-Jun-2025.)
|- ( ( ( A e. QQ \/ A = +oo ) /\ ( B e. QQ \/ B = +oo ) ) -> ( A < B <-> -. B <_ A ) )
4.6  Elementary integer functions
4.6.1  The floor and ceiling functions
Syntax	cfl 10720	Extend class notation with floor (greatest integer) function.
class |_
Syntax	cceil 10721	Extend class notation to include the ceiling function.
class ⌈
Definition	df-fl 10722*	Define the floor (greatest integer less than or equal to) function. See flval 10724 for its value, flqlelt 10729 for its basic property, and flqcl 10725 for its closure. For example, ( |_ ` ( 3 / 2 ) ) = 1 while ( |_ ` -u ( 3 / 2 ) ) = -u 2 (ex-fl 16944). 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 10723	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 10764 for its value, ceilqge 10768 and ceilqm1lt 10770 for its basic properties, and ceilqcl 10766 for its closure. For example, (⌈ ` ( 3 / 2 ) ) = 2 while (⌈ ` -u ( 3 / 2 ) ) = -u 1 (ex-ceil 16945). As described in df-fl 10722 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 10724*	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 10725	The floor (greatest integer) function yields an integer when applied to a rational (closure law). For a similar closure law for real numbers which are either rational or irrational, see flapcl 10728. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) e. ZZ )
Theorem	apbtwnz 10726*	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	flapclz 10727*	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 12985) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.)
|- ( ( A e. RR /\ A. n e. ZZ A # n ) -> ( |_ ` A ) e. ZZ )
Theorem	flapcl 10728*	The floor (greatest integer) function yields an integer when applied to a number which is either rational or irrational. (Contributed by Jim Kingdon, 20-Aug-2026.)
|- ( ( A e. QQ \/ ( A e. RR /\ A. q e. QQ A # q ) ) -> ( |_ ` A ) e. ZZ )
Theorem	flqlelt 10729	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	flaplelt 10730*	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 20-Aug-2026.)
|- ( ( A e. QQ \/ ( A e. RR /\ A. q e. QQ A # q ) ) -> ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) )
Theorem	flqcld 10731	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 10732	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) <_ A )
Theorem	flqltp1 10733	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> A < ( ( |_ ` A ) + 1 ) )
Theorem	qfraclt1 10734	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 10735	The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> 0 <_ ( A - ( |_ ` A ) ) )
Theorem	flqge 10736	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	flapge 10737*	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 \/ ( A e. RR /\ A. q e. QQ A # q ) ) /\ B e. ZZ ) -> ( B <_ A <-> B <_ ( |_ ` A ) ) )
Theorem	flqlt 10738	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	flaplt 10739*	The floor function value is less than the next integer. (Contributed by NM, 24-Feb-2005.) (Revised by Jim Kingdon, 9-Sep-2026.)
|- ( ( ( A e. QQ \/ ( A e. RR /\ A. q e. QQ A # q ) ) /\ B e. ZZ ) -> ( A < B <-> ( |_ ` A ) < B ) )
Theorem	flid 10740	An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
|- ( A e. ZZ -> ( |_ ` A ) = A )
Theorem	flqidm 10741	The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` ( |_ ` A ) ) = ( |_ ` A ) )
Theorem	flqidz 10742	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 10743	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 10744	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 10745	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 10746	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 10747	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 10748	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 10749	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 10750	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 10751	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 10752	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 10753	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 10754	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 10755	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 10756	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 10757	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 10758	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 10759	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 10760	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 10761	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	fldiv4lem1div2uz2 10762	The floor of an integer greater than 1, divided by 4 is less than or equal to the half of the integer minus 1. (Contributed by AV, 5-Jul-2021.) (Proof shortened by AV, 9-Jul-2022.)
|- ( N e. ( ZZ>= ` 2 ) -> ( |_ ` ( N / 4 ) ) <_ ( ( N - 1 ) / 2 ) )
Theorem	fldiv4lem1div2 10763	The floor of a positive integer divided by 4 is less than or equal to the half of the integer minus 1. (Contributed by AV, 9-Jul-2021.)
|- ( N e. NN -> ( |_ ` ( N / 4 ) ) <_ ( ( N - 1 ) / 2 ) )
Theorem	ceilqval 10764	The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( A e. QQ -> (⌈ ` A ) = -u ( |_ ` -u A ) )
Theorem	ceiqcl 10765	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 10766	Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> (⌈ ` A ) e. ZZ )
Theorem	ceiqge 10767	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 10768	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 10769	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 10770	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 10771	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 10772	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 10773	An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
|- ( A e. ZZ -> (⌈ ` A ) = A )
Theorem	ceilqidz 10774	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 10775	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 10776	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 10777	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 10778	Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 10777. (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 10779	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 10780	Extend class notation with the modulo operation.
class mod
Definition	df-mod 10781*	Define the modulo (remainder) operation. See modqval 10782 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10722 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 10782	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 10725 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 10783	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 10784	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 10785	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 10786	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 10787	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 10788	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 10789	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 10790	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 10791	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 10792	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 ) )
Theorem	modqdiffl 10793	The modulo operation differs from A by an integer multiple of B. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A - ( A mod B ) ) / B ) = ( |_ ` ( A / B ) ) )
Theorem	modqdifz 10794	The modulo operation differs from A by an integer multiple of B. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A - ( A mod B ) ) / B ) e. ZZ )
Theorem	modqfrac 10795	The fractional part of a number is the number modulo 1. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( A e. QQ -> ( A mod 1 ) = ( A - ( |_ ` A ) ) )
Theorem	flqmod 10796	The floor function expressed in terms of the modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) = ( A - ( A mod 1 ) ) )
Theorem	intqfrac 10797	Break a number into its integer part and its fractional part. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( A e. QQ -> A = ( ( |_ ` A ) + ( A mod 1 ) ) )
Theorem	zmod10 10798	An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( N e. ZZ -> ( N mod 1 ) = 0 )
Theorem	zmod1congr 10799	Two arbitrary integers are congruent modulo 1, see example 4 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A mod 1 ) = ( B mod 1 ) )
Theorem	modqmulnn 10800	Move a positive integer in and out of a floor in the first argument of a modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( N e. NN /\ A e. QQ /\ M e. NN ) -> ( ( N x. ( |_ ` A ) ) mod ( N x. M ) ) <_ ( ( |_ ` ( N x. A ) ) mod ( N x. M ) ) )