P. 128 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12701-12800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	0mod 12701	Special case: 0 modulo a positive real number is 0. (Contributed by Mario Carneiro, 22-Feb-2014.)
|- ( N e. RR+ -> ( 0 mod N ) = 0 )
Theorem	1mod 12702	Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- ( ( N e. RR /\ 1 < N ) -> ( 1 mod N ) = 1 )
Theorem	modabs 12703	Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)
|- ( ( ( A e. RR /\ B e. RR+ /\ C e. RR+ ) /\ B <_ C ) -> ( ( A mod B ) mod C ) = ( A mod B ) )
Theorem	modabs2 12704	Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A mod B ) mod B ) = ( A mod B ) )
Theorem	modcyc 12705	The modulo operation is periodic. (Contributed by NM, 10-Nov-2008.)
|- ( ( A e. RR /\ B e. RR+ /\ N e. ZZ ) -> ( ( A + ( N x. B ) ) mod B ) = ( A mod B ) )
Theorem	modcyc2 12706	The modulo operation is periodic. (Contributed by NM, 12-Nov-2008.)
|- ( ( A e. RR /\ B e. RR+ /\ N e. ZZ ) -> ( ( A - ( B x. N ) ) mod B ) = ( A mod B ) )
Theorem	modadd1 12707	Addition property of the modulo operation. (Contributed by NM, 12-Nov-2008.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) /\ ( A mod D ) = ( B mod D ) ) -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) )
Theorem	modaddabs 12708	Absorption law for modulo. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( ( ( A mod C ) + ( B mod C ) ) mod C ) = ( ( A + B ) mod C ) )
Theorem	modaddmod 12709	The sum of a real number modulo a positive real number and another real number equals the sum of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 13-May-2018.)
|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( ( ( A mod M ) + B ) mod M ) = ( ( A + B ) mod M ) )
Theorem	muladdmodid 12710	The sum of a positive real number less than an upper bound and the product of an integer and the upper bound is the positive real number modulo the upper bound. (Contributed by AV, 5-Jul-2020.)
|- ( ( N e. ZZ /\ M e. RR+ /\ A e. ( 0 [,) M ) ) -> ( ( ( N x. M ) + A ) mod M ) = A )
Theorem	mulp1mod1 12711	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	modmuladd 12712*	Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( ( A mod M ) = B <-> E. k e. ZZ A = ( ( k x. M ) + B ) ) )
Theorem	modmuladdim 12713*	Implication of a decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.)
|- ( ( A e. ZZ /\ M e. RR+ ) -> ( ( A mod M ) = B -> E. k e. ZZ A = ( ( k x. M ) + B ) ) )
Theorem	modmuladdnn0 12714*	Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.)
|- ( ( A e. NN0 /\ M e. RR+ ) -> ( ( A mod M ) = B -> E. k e. NN0 A = ( ( k x. M ) + B ) ) )
Theorem	negmod 12715	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 AV, 5-Jul-2020.)
|- ( ( A e. RR /\ N e. RR+ ) -> ( -u A mod N ) = ( ( N - A ) mod N ) )
Theorem	m1modnnsub1 12716	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 12717	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 12718	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 12719	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	modadd2mod 12720	The sum of a real number modulo a positive real number and another real number equals the sum of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( ( B + ( A mod M ) ) mod M ) = ( ( B + A ) mod M ) )
Theorem	modm1p1mod0 12721	If an real number modulo a positive real number equals the positive real number decreased by 1, the real number increased by 1 modulo the positive real number equals 0. (Contributed by AV, 2-Nov-2018.)
|- ( ( A e. RR /\ M e. RR+ ) -> ( ( A mod M ) = ( M - 1 ) -> ( ( A + 1 ) mod M ) = 0 ) )
Theorem	modltm1p1mod 12722	If a real number modulo a positive real number is less than the positive real number decreased by 1, the real number increased by 1 modulo the positive real number equals the real number modulo the positive real number increased by 1. (Contributed by AV, 2-Nov-2018.)
|- ( ( A e. RR /\ M e. RR+ /\ ( A mod M ) < ( M - 1 ) ) -> ( ( A + 1 ) mod M ) = ( ( A mod M ) + 1 ) )
Theorem	modmul1 12723	Multiplication property of the modulo operation. Note that the multiplier C must be an integer. (Contributed by NM, 12-Nov-2008.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ZZ /\ D e. RR+ ) /\ ( A mod D ) = ( B mod D ) ) -> ( ( A x. C ) mod D ) = ( ( B x. C ) mod D ) )
Theorem	modmul12d 12724	Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 5-Feb-2015.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> D e. ZZ )   &    |- ( ph -> E e. RR+ )   &    |- ( 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	modnegd 12725	Negation property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> ( A mod C ) = ( B mod C ) )   =>    |- ( ph -> ( -u A mod C ) = ( -u B mod C ) )
Theorem	modadd12d 12726	Additive property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> E e. RR+ )   &    |- ( ph -> ( A mod E ) = ( B mod E ) )   &    |- ( ph -> ( C mod E ) = ( D mod E ) )   =>    |- ( ph -> ( ( A + C ) mod E ) = ( ( B + D ) mod E ) )
Theorem	modsub12d 12727	Subtraction property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> E e. RR+ )   &    |- ( ph -> ( A mod E ) = ( B mod E ) )   &    |- ( ph -> ( C mod E ) = ( D mod E ) )   =>    |- ( ph -> ( ( A - C ) mod E ) = ( ( B - D ) mod E ) )
Theorem	modsubmod 12728	The difference of a real number modulo a positive real number and another real number equals the difference of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( ( ( A mod M ) - B ) mod M ) = ( ( A - B ) mod M ) )
Theorem	modsubmodmod 12729	The difference of a real number modulo a positive real number and another real number modulo this positive real number equals the difference of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( ( ( A mod M ) - ( B mod M ) ) mod M ) = ( ( A - B ) mod M ) )
Theorem	2txmodxeq0 12730	Two times a positive real number modulo the real number is zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
|- ( X e. RR+ -> ( ( 2 x. X ) mod X ) = 0 )
Theorem	2submod 12731	If a real number is between a positive real number and twice the positive real number, the real number modulo the positive real number equals the real number minus the positive real number. (Contributed by Alexander van der Vekens, 13-May-2018.)
|- ( ( ( A e. RR /\ B e. RR+ ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A mod B ) = ( A - B ) )
Theorem	modifeq2int 12732	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 12733	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 12734	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	modmulmod 12735	The product of a real number modulo a positive real number and an integer equals the product of the real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
|- ( ( A e. RR /\ B e. ZZ /\ M e. RR+ ) -> ( ( ( A mod M ) x. B ) mod M ) = ( ( A x. B ) mod M ) )
Theorem	modmulmodr 12736	The product of an integer and a real number modulo a positive real number equals the product of the integer and the real number modulo the positive real number. (Contributed by Alexander van der Vekens, 9-Jul-2021.)
|- ( ( A e. ZZ /\ B e. RR /\ M e. RR+ ) -> ( ( A x. ( B mod M ) ) mod M ) = ( ( A x. B ) mod M ) )
Theorem	modaddmulmod 12737	The sum of a real number and the product of a second real number modulo a positive real number and an integer equals the sum of the real number and the product of the other real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
|- ( ( ( A e. RR /\ B e. RR /\ C e. ZZ ) /\ M e. RR+ ) -> ( ( A + ( ( B mod M ) x. C ) ) mod M ) = ( ( A + ( B x. C ) ) mod M ) )
Theorem	moddi 12738	Distribute multiplication over a modulo operation. (Contributed by NM, 11-Nov-2008.)
|- ( ( A e. RR+ /\ B e. RR /\ C e. RR+ ) -> ( A x. ( B mod C ) ) = ( ( A x. B ) mod ( A x. C ) ) )
Theorem	modsubdir 12739	Distribute the modulo operation over a subtraction. (Contributed by NM, 30-Dec-2008.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR+ ) -> ( ( B mod C ) <_ ( A mod C ) <-> ( ( A - B ) mod C ) = ( ( A mod C ) - ( B mod C ) ) ) )
Theorem	modeqmodmin 12740	A real number equals the difference of the real number and a positive real number modulo the positive real number. (Contributed by AV, 3-Nov-2018.)
|- ( ( A e. RR /\ M e. RR+ ) -> ( A mod M ) = ( ( A - M ) mod M ) )
Theorem	modirr 12741	A number modulo an irrational multiple of it is nonzero. (Contributed by NM, 11-Nov-2008.)
|- ( ( A e. RR /\ B e. RR+ /\ ( A / B ) e. ( RR \ QQ ) ) -> ( A mod B ) =/= 0 )
Theorem	modfzo0difsn 12742*	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 12743	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 } ) )
Theorem	modlteq 12744	Two nonnegative integers less than the modulus are equal iff they are equal modulo the modulus. (Contributed by AV, 14-Mar-2021.)
|- ( ( I e. ( 0..^ N ) /\ J e. ( 0..^ N ) ) -> ( ( I mod N ) = ( J mod N ) <-> I = J ) )
Theorem	addmodlteq 12745	Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. A much shorter proof exists if the "divides" relation || can be used, see addmodlteqALT 15047. (Contributed by AV, 20-Mar-2021.)
|- ( ( I e. ( 0..^ N ) /\ J e. ( 0..^ N ) /\ S e. ZZ ) -> ( ( ( I + S ) mod N ) = ( ( J + S ) mod N ) <-> I = J ) )
5.6.3  Miscellaneous theorems about integers
Theorem	om2uz0i 12746*	The mapping G is a one-to-one mapping from om onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number C (normally 0 for the upper integers NN0 or 1 for the upper integers NN), 1 maps to C + 1, etc. This theorem shows the value of G at ordinal natural number zero. (This series of theorems generalizes an earlier series for NN0 contributed by Raph Levien, 10-Apr-2004.) (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   =>    |- ( G ` (/) ) = C
Theorem	om2uzsuci 12747*	The value of G (see om2uz0i 12746) at a successor. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   =>    |- ( A e. om -> ( G ` suc A ) = ( ( G ` A ) + 1 ) )
Theorem	om2uzuzi 12748*	The value G (see om2uz0i 12746) at an ordinal natural number is in the upper integers. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   =>    |- ( A e. om -> ( G ` A ) e. ( ZZ>= ` C ) )
Theorem	om2uzlti 12749*	Less-than relation for G (see om2uz0i 12746). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   =>    |- ( ( A e. om /\ B e. om ) -> ( A e. B -> ( G ` A ) < ( G ` B ) ) )
Theorem	om2uzlt2i 12750*	The mapping G (see om2uz0i 12746) preserves order. (Contributed by NM, 4-May-2005.) (Revised by Mario Carneiro, 13-Sep-2013.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   =>    |- ( ( A e. om /\ B e. om ) -> ( A e. B <-> ( G ` A ) < ( G ` B ) ) )
Theorem	om2uzrani 12751*	Range of G (see om2uz0i 12746). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   =>    |- ran G = ( ZZ>= ` C )
Theorem	om2uzf1oi 12752*	G (see om2uz0i 12746) is a one-to-one onto mapping. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   =>    |- G : om -1-1-onto-> ( ZZ>= ` C )
Theorem	om2uzisoi 12753*	G (see om2uz0i 12746) is an isomorphism from natural ordinals to upper integers. (Contributed by NM, 9-Oct-2008.) (Revised by Mario Carneiro, 13-Sep-2013.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   =>    |- G Isom _E , < ( om , ( ZZ>= ` C ) )
Theorem	om2uzoi 12754*	An alternative definition of G in terms of df-oi 8415. (Contributed by Mario Carneiro, 2-Jun-2015.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   =>    |- G = OrdIso ( < , ( ZZ>= ` C ) )
Theorem	om2uzrdg 12755*	A helper lemma for the value of a recursive definition generator on upper integers (typically either NN or NN0) with characteristic function F ( x , y ) and initial value A. Normally F is a function on the partition, and A is a member of the partition. See also comment in om2uz0i 12746. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   &    |- A e. _V   &    |- R = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om )   =>    |- ( B e. om -> ( R ` B ) = <. ( G ` B ) , ( 2nd ` ( R ` B ) ) >. )
Theorem	uzrdglem 12756*	A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   &    |- A e. _V   &    |- R = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om )   =>    |- ( B e. ( ZZ>= ` C ) -> <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. ran R )
Theorem	uzrdgfni 12757*	The recursive definition generator on upper integers is a function. See comment in om2uzrdg 12755. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 4-May-2015.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   &    |- A e. _V   &    |- R = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om )   &    |- S = ran R   =>    |- S Fn ( ZZ>= ` C )
Theorem	uzrdg0i 12758*	Initial value of a recursive definition generator on upper integers. See comment in om2uzrdg 12755. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   &    |- A e. _V   &    |- R = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om )   &    |- S = ran R   =>    |- ( S ` C ) = A
Theorem	uzrdgsuci 12759*	Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 12755. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
|- C e. ZZ   &    |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , C ) |` om )   &    |- A e. _V   &    |- R = ( rec ( ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. ) |` om )   &    |- S = ran R   =>    |- ( B e. ( ZZ>= ` C ) -> ( S ` ( B + 1 ) ) = ( B F ( S ` B ) ) )
Theorem	ltweuz 12760	< is a well-founded relation on any sequence of upper integers. (Contributed by Andrew Salmon, 13-Nov-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
|- < We ( ZZ>= ` A )
Theorem	ltwenn 12761	Less than well-orders the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
|- < We NN
Theorem	ltwefz 12762	Less than well-orders a set of finite integers. (Contributed by Scott Fenton, 8-Aug-2013.)
|- < We ( M ... N )
Theorem	uzenom 12763	An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> Z ~~ om )
Theorem	uzinf 12764	An upper integer set is infinite. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> -. Z e. Fin )
Theorem	nnnfi 12765	The set of positive integers is infinite. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- -. NN e. Fin
Theorem	uzrdgxfr 12766*	Transfer the value of the recursive sequence builder from one base to another. (Contributed by Mario Carneiro, 1-Apr-2014.)
|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , A ) |` om )   &    |- H = ( rec ( ( x e. _V |-> ( x + 1 ) ) , B ) |` om )   &    |- A e. ZZ   &    |- B e. ZZ   =>    |- ( N e. om -> ( G ` N ) = ( ( H ` N ) + ( A - B ) ) )
Theorem	fzennn 12767	The cardinality of a finite set of sequential integers. (See om2uz0i 12746 for a description of the hypothesis.) (Contributed by Mario Carneiro, 12-Feb-2013.) (Revised by Mario Carneiro, 7-Mar-2014.)
|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om )   =>    |- ( N e. NN0 -> ( 1 ... N ) ~~ ( `' G ` N ) )
Theorem	fzen2 12768	The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Mario Carneiro, 13-Feb-2014.)
|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om )   =>    |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( `' G ` ( ( N + 1 ) - M ) ) )
Theorem	cardfz 12769	The cardinality of a finite set of sequential integers. (See om2uz0i 12746 for a description of the hypothesis.) (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 15-Sep-2013.)
|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om )   =>    |- ( N e. NN0 -> ( card ` ( 1 ... N ) ) = ( `' G ` N ) )
Theorem	hashgf1o 12770	G maps om one-to-one onto NN0. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 13-Sep-2013.)
|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om )   =>    |- G : om -1-1-onto-> NN0
Theorem	fzfi 12771	A finite interval of integers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
|- ( M ... N ) e. Fin
Theorem	fzfid 12772	Commonly used special case of fzfi 12771. (Contributed by Mario Carneiro, 25-May-2014.)
|- ( ph -> ( M ... N ) e. Fin )
Theorem	fzofi 12773	Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015.)
|- ( M..^ N ) e. Fin
Theorem	fsequb 12774*	The values of a finite real sequence have an upper bound. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
|- ( A. k e. ( M ... N ) ( F ` k ) e. RR -> E. x e. RR A. k e. ( M ... N ) ( F ` k ) < x )
Theorem	fsequb2 12775*	The values of a finite real sequence have an upper bound. (Contributed by NM, 20-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
|- ( F : ( M ... N ) --> RR -> E. x e. RR A. y e. ran F y <_ x )
Theorem	fseqsupcl 12776	The values of a finite real sequence have a supremum. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ( N e. ( ZZ>= ` M ) /\ F : ( M ... N ) --> RR ) -> sup ( ran F , RR , < ) e. RR )
Theorem	fseqsupubi 12777	The values of a finite real sequence are bounded by their supremum. (Contributed by NM, 20-Sep-2005.)
|- ( ( K e. ( M ... N ) /\ F : ( M ... N ) --> RR ) -> ( F ` K ) <_ sup ( ran F , RR , < ) )
Theorem	nn0ennn 12778	The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
|- NN0 ~~ NN
Theorem	nnenom 12779	The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of finite ordinal numbers). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
|- NN ~~ om
Theorem	nnct 12780	NN is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
|- NN ~<_ om
Theorem	uzindi 12781*	Indirect strong induction on the upper integers. (Contributed by Stefan O'Rear, 25-Aug-2015.)
|- ( ph -> A e. V )   &    |- ( ph -> T e. ( ZZ>= ` L ) )   &    |- ( ( ph /\ R e. ( L ... T ) /\ A. y ( S e. ( L..^ R ) -> ch ) ) -> ps )   &    |- ( x = y -> ( ps <-> ch ) )   &    |- ( x = A -> ( ps <-> th ) )   &    |- ( x = y -> R = S )   &    |- ( x = A -> R = T )   =>    |- ( ph -> th )
Theorem	axdc4uzlem 12782*	Lemma for axdc4uz 12783. (Contributed by Mario Carneiro, 8-Jan-2014.) (Revised by Mario Carneiro, 26-Dec-2014.)
|- M e. ZZ   &    |- Z = ( ZZ>= ` M )   &    |- A e. _V   &    |- G = ( rec ( ( y e. _V |-> ( y + 1 ) ) , M ) |` om )   &    |- H = ( n e. om , x e. A |-> ( ( G ` n ) F x ) )   =>    |- ( ( C e. A /\ F : ( Z X. A ) --> ( ~P A \ { (/) } ) ) -> E. g ( g : Z --> A /\ ( g ` M ) = C /\ A. k e. Z ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) ) )
Theorem	axdc4uz 12783*	A version of axdc4 9278 that works on an upper set of integers instead of om. (Contributed by Mario Carneiro, 8-Jan-2014.)
|- M e. ZZ   &    |- Z = ( ZZ>= ` M )   =>    |- ( ( A e. V /\ C e. A /\ F : ( Z X. A ) --> ( ~P A \ { (/) } ) ) -> E. g ( g : Z --> A /\ ( g ` M ) = C /\ A. k e. Z ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) ) )
Theorem	ssnn0fi 12784*	A subset of the nonnegative integers is finite if and only if there is a nonnegative integer so that all integers greater than this integer are not contained in the subset. (Contributed by AV, 3-Oct-2019.)
|- ( S C_ NN0 -> ( S e. Fin <-> E. s e. NN0 A. x e. NN0 ( s < x -> x e/ S ) ) )
Theorem	rabssnn0fi 12785*	A subset of the nonnegative integers defined by a restricted class abstraction is finite if there is a nonnegative integer so that for all integers greater than this integer the condition of the class abstraction is not fulfilled. (Contributed by AV, 3-Oct-2019.)
|- ( { x e. NN0 | ph } e. Fin <-> E. s e. NN0 A. x e. NN0 ( s < x -> -. ph ) )
5.6.4  Strong induction over upper sets of integers
Theorem	uzsinds 12786*	Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = N -> ( ph <-> ch ) )   &    |- ( x e. ( ZZ>= ` M ) -> ( A. y e. ( M ... ( x - 1 ) ) ps -> ph ) )   =>    |- ( N e. ( ZZ>= ` M ) -> ch )
Theorem	nnsinds 12787*	Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = N -> ( ph <-> ch ) )   &    |- ( x e. NN -> ( A. y e. ( 1 ... ( x - 1 ) ) ps -> ph ) )   =>    |- ( N e. NN -> ch )
Theorem	nn0sinds 12788*	Strong (or "total") induction principle over the nonnegative integers. (Contributed by Scott Fenton, 16-May-2014.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = N -> ( ph <-> ch ) )   &    |- ( x e. NN0 -> ( A. y e. ( 0 ... ( x - 1 ) ) ps -> ph ) )   =>    |- ( N e. NN0 -> ch )
5.6.5  Finitely supported functions over the nonnegative integers
Theorem	fsuppmapnn0fiublem 12789*	Lemma for fsuppmapnn0fiub 12790 and fsuppmapnn0fiubex 12792. (Contributed by AV, 2-Oct-2019.)
|- U = U_ f e. M ( f supp Z )   &    |- S = sup ( U , RR , < )   =>    |- ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) -> ( ( A. f e. M f finSupp Z /\ U =/= (/) ) -> S e. NN0 ) )
Theorem	fsuppmapnn0fiub 12790*	If all functions of a finite set of functions over the nonnegative integers are finitely supported, then the support of all these functions is contained in a finite set of sequential integers starting at 0 and ending with the supremum of the union of the support of these functions. (Contributed by AV, 2-Oct-2019.) (Proof shortened by JJ, 2-Aug-2021.)
|- U = U_ f e. M ( f supp Z )   &    |- S = sup ( U , RR , < )   =>    |- ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) -> ( ( A. f e. M f finSupp Z /\ U =/= (/) ) -> A. f e. M ( f supp Z ) C_ ( 0 ... S ) ) )
Theorem	fsuppmapnn0fiubOLD 12791*	Obsolete proof of fsuppmapnn0fiub 12790 as of 2-Aug-2021. (Contributed by AV, 2-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
|- U = U_ f e. M ( f supp Z )   &    |- S = sup ( U , RR , < )   =>    |- ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) -> ( ( A. f e. M f finSupp Z /\ U =/= (/) ) -> A. f e. M ( f supp Z ) C_ ( 0 ... S ) ) )
Theorem	fsuppmapnn0fiubex 12792*	If all functions of a finite set of functions over the nonnegative integers are finitely supported, then the support of all these functions is contained in a finite set of sequential integers starting at 0. (Contributed by AV, 2-Oct-2019.)
|- ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) -> ( A. f e. M f finSupp Z -> E. m e. NN0 A. f e. M ( f supp Z ) C_ ( 0 ... m ) ) )
Theorem	fsuppmapnn0fiub0 12793*	If all functions of a finite set of functions over the nonnegative integers are finitely supported, then all these functions are zero for all integers greater than a fixed integer. (Contributed by AV, 3-Oct-2019.)
|- ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) -> ( A. f e. M f finSupp Z -> E. m e. NN0 A. f e. M A. x e. NN0 ( m < x -> ( f ` x ) = Z ) ) )
Theorem	suppssfz 12794*	Condition for a function over the nonnegative integers to have a support contained in a finite set of sequential integers. (Contributed by AV, 9-Oct-2019.)
|- ( ph -> Z e. V )   &    |- ( ph -> F e. ( B ^m NN0 ) )   &    |- ( ph -> S e. NN0 )   &    |- ( ph -> A. x e. NN0 ( S < x -> ( F ` x ) = Z ) )   =>    |- ( ph -> ( F supp Z ) C_ ( 0 ... S ) )
Theorem	fsuppmapnn0ub 12795*	If a function over the nonnegative integers is finitely supported, then there is an upper bound for the arguments resulting in nonzero values. (Contributed by AV, 6-Oct-2019.)
|- ( ( F e. ( R ^m NN0 ) /\ Z e. V ) -> ( F finSupp Z -> E. m e. NN0 A. x e. NN0 ( m < x -> ( F ` x ) = Z ) ) )
Theorem	fsuppmapnn0fz 12796*	If a function over the nonnegative integers is finitely supported, then there is an upper bound for a finite set of sequential integers containing the support of the function. (Contributed by AV, 30-Sep-2019.) (Proof shortened by AV, 6-Oct-2019.)
|- ( ( F e. ( R ^m NN0 ) /\ Z e. V ) -> ( F finSupp Z -> E. m e. NN0 ( F supp Z ) C_ ( 0 ... m ) ) )
Theorem	mptnn0fsupp 12797*	A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 23-Dec-2019.)
|- ( ph -> .0. e. V )   &    |- ( ( ph /\ k e. NN0 ) -> C e. B )   &    |- ( ph -> E. s e. NN0 A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) )   =>    |- ( ph -> ( k e. NN0 |-> C ) finSupp .0. )
Theorem	mptnn0fsuppd 12798*	A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 2-Dec-2019.) (Revised by AV, 23-Dec-2019.)
|- ( ph -> .0. e. V )   &    |- ( ( ph /\ k e. NN0 ) -> C e. B )   &    |- ( k = x -> C = D )   &    |- ( ph -> E. s e. NN0 A. x e. NN0 ( s < x -> D = .0. ) )   =>    |- ( ph -> ( k e. NN0 |-> C ) finSupp .0. )
Theorem	mptnn0fsuppr 12799*	A finitely supported mapping from the nonnegative integers fulfills certain conditions. (Contributed by AV, 3-Nov-2019.) (Revised by AV, 23-Dec-2019.)
|- ( ph -> .0. e. V )   &    |- ( ( ph /\ k e. NN0 ) -> C e. B )   &    |- ( ph -> ( k e. NN0 |-> C ) finSupp .0. )   =>    |- ( ph -> E. s e. NN0 A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) )
Theorem	f13idfv 12800	A one-to-one function with the domain { 0, 1 ,2 } in terms of function values. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
|- A = ( 0 ... 2 )   =>    |- ( F : A -1-1-> B <-> ( F : A --> B /\ ( ( F ` 0 ) =/= ( F ` 1 ) /\ ( F ` 0 ) =/= ( F ` 2 ) /\ ( F ` 1 ) =/= ( F ` 2 ) ) ) )