P. 102 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10101-10200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	flqaddz 10101	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 10102	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 10103	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 10104	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 10105	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 10106	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 10107	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 10108	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 10109	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 10110	The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( A e. QQ -> (⌈ ` A ) = -u ( |_ ` -u A ) )
Theorem	ceiqcl 10111	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 10112	Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> (⌈ ` A ) e. ZZ )
Theorem	ceiqge 10113	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 10114	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 10115	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 10116	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 10117	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 10118	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 10119	An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
|- ( A e. ZZ -> (⌈ ` A ) = A )
Theorem	ceilqidz 10120	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 10121	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 10122	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 10123	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 10124	Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 10123. (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 10125	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 10126	Extend class notation with the modulo operation.
class mod
Definition	df-mod 10127*	Define the modulo (remainder) operation. See modqval 10128 for its value. For example, ( 5 mod 3 ) = 2 and ( -u 7 mod 2 ) = 1. As with df-fl 10074 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 10128	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 10077 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 10129	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 10130	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 10131	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 10132	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 10133	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 10134	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 10135	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 10136	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 10137	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 10138	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 10139	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 10140	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 10141	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 10142	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 10143	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 10144	An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( N e. ZZ -> ( N mod 1 ) = 0 )
Theorem	zmod1congr 10145	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 10146	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 10147	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 10148	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 10149	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 10150	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 10151	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 10152	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 10153	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 10154	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 10155	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 10156	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 10157	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 10158	Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
|- ( M e. ( 0..^ N ) -> ( M mod N ) = M )
Theorem	q0mod 10159	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 10160	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 10161	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 ) )
Theorem	modqabs2 10162	Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) mod B ) = ( A mod B ) )
Theorem	modqcyc 10163	The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( ( A + ( N x. B ) ) mod B ) = ( A mod B ) )
Theorem	modqcyc2 10164	The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.)
|- ( ( ( A e. QQ /\ N e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( ( A - ( B x. N ) ) mod B ) = ( A mod B ) )
Theorem	modqadd1 10165	Addition property of the modulo operation. (Contributed by Jim Kingdon, 22-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> C e. QQ )   &    |- ( ph -> D e. QQ )   &    |- ( ph -> 0 < D )   &    |- ( ph -> ( A mod D ) = ( B mod D ) )   =>    |- ( ph -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) )
Theorem	modqaddabs 10166	Absorption law for modulo. (Contributed by Jim Kingdon, 22-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( ( A mod C ) + ( B mod C ) ) mod C ) = ( ( A + B ) mod C ) )
Theorem	modqaddmod 10167	The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the same modulus. (Contributed by Jim Kingdon, 23-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) + B ) mod M ) = ( ( A + B ) mod M ) )
Theorem	mulqaddmodid 10168	The sum of a positive rational number less than an upper bound and the product of an integer and the upper bound is the positive rational number modulo the upper bound. (Contributed by Jim Kingdon, 23-Oct-2021.)
|- ( ( ( N e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( ( ( N x. M ) + A ) mod M ) = A )
Theorem	mulp1mod1 10169	The product of an integer and an integer greater than 1 increased by 1 is 1 modulo the integer greater than 1. (Contributed by AV, 15-Jul-2021.)
|- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) + 1 ) mod N ) = 1 )
Theorem	modqmuladd 10170*	Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> B e. ( 0 [,) M ) )   &    |- ( ph -> M e. QQ )   &    |- ( ph -> 0 < M )   =>    |- ( ph -> ( ( A mod M ) = B <-> E. k e. ZZ A = ( ( k x. M ) + B ) ) )
Theorem	modqmuladdim 10171*	Implication of a decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
|- ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = B -> E. k e. ZZ A = ( ( k x. M ) + B ) ) )
Theorem	modqmuladdnn0 10172*	Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
|- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = B -> E. k e. NN0 A = ( ( k x. M ) + B ) ) )
Theorem	qnegmod 10173	The negation of a number modulo a positive number is equal to the difference of the modulus and the number modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( -u A mod N ) = ( ( N - A ) mod N ) )
Theorem	m1modnnsub1 10174	Minus one modulo a positive integer is equal to the integer minus one. (Contributed by AV, 14-Jul-2021.)
|- ( M e. NN -> ( -u 1 mod M ) = ( M - 1 ) )
Theorem	m1modge3gt1 10175	Minus one modulo an integer greater than two is greater than one. (Contributed by AV, 14-Jul-2021.)
|- ( M e. ( ZZ>= ` 3 ) -> 1 < ( -u 1 mod M ) )
Theorem	addmodid 10176	The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by Alexander van der Vekens, 30-Oct-2018.) (Proof shortened by AV, 5-Jul-2020.)
|- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> ( ( M + A ) mod M ) = A )
Theorem	addmodidr 10177	The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by AV, 19-Mar-2021.)
|- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> ( ( A + M ) mod M ) = A )
Theorem	modqadd2mod 10178	The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B + ( A mod M ) ) mod M ) = ( ( B + A ) mod M ) )
Theorem	modqm1p1mod0 10179	If a number modulo a modulus equals the modulus decreased by 1, the first number increased by 1 modulo the modulus equals 0. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = ( M - 1 ) -> ( ( A + 1 ) mod M ) = 0 ) )
Theorem	modqltm1p1mod 10180	If a number modulo a modulus is less than the modulus decreased by 1, the first number increased by 1 modulo the modulus equals the first number modulo the modulus, increased by 1. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A + 1 ) mod M ) = ( ( A mod M ) + 1 ) )
Theorem	modqmul1 10181	Multiplication property of the modulo operation. Note that the multiplier C must be an integer. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> D e. QQ )   &    |- ( ph -> 0 < D )   &    |- ( ph -> ( A mod D ) = ( B mod D ) )   =>    |- ( ph -> ( ( A x. C ) mod D ) = ( ( B x. C ) mod D ) )
Theorem	modqmul12d 10182	Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> D e. ZZ )   &    |- ( ph -> E e. QQ )   &    |- ( ph -> 0 < E )   &    |- ( ph -> ( A mod E ) = ( B mod E ) )   &    |- ( ph -> ( C mod E ) = ( D mod E ) )   =>    |- ( ph -> ( ( A x. C ) mod E ) = ( ( B x. D ) mod E ) )
Theorem	modqnegd 10183	Negation property of the modulo operation. (Contributed by Jim Kingdon, 24-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> C e. QQ )   &    |- ( ph -> 0 < C )   &    |- ( ph -> ( A mod C ) = ( B mod C ) )   =>    |- ( ph -> ( -u A mod C ) = ( -u B mod C ) )
Theorem	modqadd12d 10184	Additive property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> C e. QQ )   &    |- ( ph -> D e. QQ )   &    |- ( ph -> E e. QQ )   &    |- ( ph -> 0 < E )   &    |- ( ph -> ( A mod E ) = ( B mod E ) )   &    |- ( ph -> ( C mod E ) = ( D mod E ) )   =>    |- ( ph -> ( ( A + C ) mod E ) = ( ( B + D ) mod E ) )
Theorem	modqsub12d 10185	Subtraction property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> C e. QQ )   &    |- ( ph -> D e. QQ )   &    |- ( ph -> E e. QQ )   &    |- ( ph -> 0 < E )   &    |- ( ph -> ( A mod E ) = ( B mod E ) )   &    |- ( ph -> ( C mod E ) = ( D mod E ) )   =>    |- ( ph -> ( ( A - C ) mod E ) = ( ( B - D ) mod E ) )
Theorem	modqsubmod 10186	The difference of a number modulo a modulus and another number equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) - B ) mod M ) = ( ( A - B ) mod M ) )
Theorem	modqsubmodmod 10187	The difference of a number modulo a modulus and another number modulo the same modulus equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) - ( B mod M ) ) mod M ) = ( ( A - B ) mod M ) )
Theorem	q2txmodxeq0 10188	Two times a positive number modulo the number is zero. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ( X e. QQ /\ 0 < X ) -> ( ( 2 x. X ) mod X ) = 0 )
Theorem	q2submod 10189	If a number is between a modulus and twice the modulus, the first number modulo the modulus equals the first number minus the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A mod B ) = ( A - B ) )
Theorem	modifeq2int 10190	If a nonnegative integer is less than twice a positive integer, the nonnegative integer modulo the positive integer equals the nonnegative integer or the nonnegative integer minus the positive integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
|- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) )
Theorem	modaddmodup 10191	The sum of an integer modulo a positive integer and another integer minus the positive integer equals the sum of the two integers modulo the positive integer if the other integer is in the upper part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
|- ( ( A e. ZZ /\ M e. NN ) -> ( B e. ( ( M - ( A mod M ) )..^ M ) -> ( ( B + ( A mod M ) ) - M ) = ( ( B + A ) mod M ) ) )
Theorem	modaddmodlo 10192	The sum of an integer modulo a positive integer and another integer equals the sum of the two integers modulo the positive integer if the other integer is in the lower part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
|- ( ( A e. ZZ /\ M e. NN ) -> ( B e. ( 0..^ ( M - ( A mod M ) ) ) -> ( B + ( A mod M ) ) = ( ( B + A ) mod M ) ) )
Theorem	modqmulmod 10193	The product of a rational number modulo a modulus and an integer equals the product of the rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) x. B ) mod M ) = ( ( A x. B ) mod M ) )
Theorem	modqmulmodr 10194	The product of an integer and a rational number modulo a modulus equals the product of the integer and the rational number modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
|- ( ( ( A e. ZZ /\ B e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A x. ( B mod M ) ) mod M ) = ( ( A x. B ) mod M ) )
Theorem	modqaddmulmod 10195	The sum of a rational number and the product of a second rational number modulo a modulus and an integer equals the sum of the rational number and the product of the other rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A + ( ( B mod M ) x. C ) ) mod M ) = ( ( A + ( B x. C ) ) mod M ) )
Theorem	modqdi 10196	Distribute multiplication over a modulo operation. (Contributed by Jim Kingdon, 26-Oct-2021.)
|- ( ( ( A e. QQ /\ 0 < A ) /\ B e. QQ /\ ( C e. QQ /\ 0 < C ) ) -> ( A x. ( B mod C ) ) = ( ( A x. B ) mod ( A x. C ) ) )
Theorem	modqsubdir 10197	Distribute the modulo operation over a subtraction. (Contributed by Jim Kingdon, 26-Oct-2021.)
|- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( B mod C ) <_ ( A mod C ) <-> ( ( A - B ) mod C ) = ( ( A mod C ) - ( B mod C ) ) ) )
Theorem	modqeqmodmin 10198	A rational number equals the difference of the rational number and a modulus modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
|- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( A mod M ) = ( ( A - M ) mod M ) )
Theorem	modfzo0difsn 10199*	For a number within a half-open range of nonnegative integers with one excluded integer there is a positive integer so that the number is equal to the sum of the positive integer and the excluded integer modulo the upper bound of the range. (Contributed by AV, 19-Mar-2021.)
|- ( ( J e. ( 0..^ N ) /\ K e. ( ( 0..^ N ) \ { J } ) ) -> E. i e. ( 1..^ N ) K = ( ( i + J ) mod N ) )
Theorem	modsumfzodifsn 10200	The sum of a number within a half-open range of positive integers is an element of the corresponding open range of nonnegative integers with one excluded integer modulo the excluded integer. (Contributed by AV, 19-Mar-2021.)
|- ( ( J e. ( 0..^ N ) /\ K e. ( 1..^ N ) ) -> ( ( K + J ) mod N ) e. ( ( 0..^ N ) \ { J } ) )