P. 105 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ioom 10401*	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 10402	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 10403	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 10404	Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
|- RR+ = ( 0 (,) +oo )
Theorem	elicod 10405	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 10406	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 10407	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 10408	"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 10409	Extend class notation with floor (greatest integer) function.
class |_
Syntax	cceil 10410	Extend class notation to include the ceiling function.
class ⌈
Definition	df-fl 10411*	Define the floor (greatest integer less than or equal to) function. See flval 10413 for its value, flqlelt 10417 for its basic property, and flqcl 10414 for its closure. For example, ( |_ ` ( 3 / 2 ) ) = 1 while ( |_ ` -u ( 3 / 2 ) ) = -u 2 (ex-fl 15594). 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 10412	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 10449 for its value, ceilqge 10453 and ceilqm1lt 10455 for its basic properties, and ceilqcl 10451 for its closure. For example, (⌈ ` ( 3 / 2 ) ) = 2 while (⌈ ` -u ( 3 / 2 ) ) = -u 1 (ex-ceil 15595). As described in df-fl 10411 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 10413*	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 10414	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 10416. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) e. ZZ )
Theorem	apbtwnz 10415*	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 10416*	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 12444) 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 10417	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 10418	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 10419	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) <_ A )
Theorem	flqltp1 10420	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> A < ( ( |_ ` A ) + 1 ) )
Theorem	qfraclt1 10421	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 10422	The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> 0 <_ ( A - ( |_ ` A ) ) )
Theorem	flqge 10423	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 10424	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 10425	An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
|- ( A e. ZZ -> ( |_ ` A ) = A )
Theorem	flqidm 10426	The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` ( |_ ` A ) ) = ( |_ ` A ) )
Theorem	flqidz 10427	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 10428	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 10429	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 10430	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 10431	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 10432	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 10433	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 10434	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 10435	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 10436	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 10437	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 10438	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 10439	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 10440	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 10441	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 10442	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 10443	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 10444	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 10445	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 10446	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 10447	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 10448	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 10449	The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( A e. QQ -> (⌈ ` A ) = -u ( |_ ` -u A ) )
Theorem	ceiqcl 10450	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 10451	Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> (⌈ ` A ) e. ZZ )
Theorem	ceiqge 10452	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 10453	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 10454	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 10455	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 10456	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 10457	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 10458	An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
|- ( A e. ZZ -> (⌈ ` A ) = A )
Theorem	ceilqidz 10459	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 10460	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 10461	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 10462	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 10463	Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 10462. (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 10464	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 10465	Extend class notation with the modulo operation.
class mod
Definition	df-mod 10466*	Define the modulo (remainder) operation. See modqval 10467 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10411 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 10467	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 10414 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 10468	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 10469	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 10470	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 10471	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 10472	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 10473	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 10474	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 10475	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 10476	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 10477	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 10478	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 10479	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 10480	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 10481	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 10482	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 10483	An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( N e. ZZ -> ( N mod 1 ) = 0 )
Theorem	zmod1congr 10484	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 10485	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 ) ) )
Theorem	modqvalp1 10486	The value of the modulo operation (expressed with sum of denominator and nominator). (Contributed by Jim Kingdon, 20-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A + B ) - ( ( ( |_ ` ( A / B ) ) + 1 ) x. B ) ) = ( A mod B ) )
Theorem	zmodcl 10487	Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.)
|- ( ( A e. ZZ /\ B e. NN ) -> ( A mod B ) e. NN0 )
Theorem	zmodcld 10488	Closure law for the modulo operation restricted to integers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. NN )   =>    |- ( ph -> ( A mod B ) e. NN0 )
Theorem	zmodfz 10489	An integer mod B lies in the first B nonnegative integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( ( A e. ZZ /\ B e. NN ) -> ( A mod B ) e. ( 0 ... ( B - 1 ) ) )
Theorem	zmodfzo 10490	An integer mod B lies in the first B nonnegative integers. (Contributed by Stefan O'Rear, 6-Sep-2015.)
|- ( ( A e. ZZ /\ B e. NN ) -> ( A mod B ) e. ( 0..^ B ) )
Theorem	zmodfzp1 10491	An integer mod B lies in the first B + 1 nonnegative integers. (Contributed by AV, 27-Oct-2018.)
|- ( ( A e. ZZ /\ B e. NN ) -> ( A mod B ) e. ( 0 ... B ) )
Theorem	modqid 10492	Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = A )
Theorem	modqid0 10493	A positive real number modulo itself is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( N e. QQ /\ 0 < N ) -> ( N mod N ) = 0 )
Theorem	modqid2 10494	Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = A <-> ( 0 <_ A /\ A < B ) ) )
Theorem	zmodid2 10495	Identity law for modulo restricted to integers. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( M e. ZZ /\ N e. NN ) -> ( ( M mod N ) = M <-> M e. ( 0 ... ( N - 1 ) ) ) )
Theorem	zmodidfzo 10496	Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
|- ( ( M e. ZZ /\ N e. NN ) -> ( ( M mod N ) = M <-> M e. ( 0..^ N ) ) )
Theorem	zmodidfzoimp 10497	Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
|- ( M e. ( 0..^ N ) -> ( M mod N ) = M )
Theorem	q0mod 10498	Special case: 0 modulo a positive real number is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( N e. QQ /\ 0 < N ) -> ( 0 mod N ) = 0 )
Theorem	q1mod 10499	Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( N e. QQ /\ 1 < N ) -> ( 1 mod N ) = 1 )
Theorem	modqabs 10500	Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> 0 < B )   &    |- ( ph -> C e. QQ )   &    |- ( ph -> B <_ C )   =>    |- ( ph -> ( ( A mod B ) mod C ) = ( A mod B ) )