P. 107 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10601-10700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	modqmuladdim 10601*	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 10602*	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 10603	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 10604	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 10605	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 10606	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 10607	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 10608	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 10609	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 10610	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 10611	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 10612	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 10613	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 10614	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 10615	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 10616	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 10617	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 10618	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 10619	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 10620	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 10621	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 10622	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 10623	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 10624	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 10625	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 10626	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 10627	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 10628	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 10629*	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 10630	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 10631	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 10632	Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. (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 ) )
4.6.3  Miscellaneous theorems about integers
Theorem	frec2uz0d 10633*	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. (Contributed by Jim Kingdon, 16-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> ( G ` (/) ) = C )
Theorem	frec2uzzd 10634*	The value of G (see frec2uz0d 10633) is an integer. (Contributed by Jim Kingdon, 16-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. om )   =>    |- ( ph -> ( G ` A ) e. ZZ )
Theorem	frec2uzsucd 10635*	The value of G (see frec2uz0d 10633) at a successor. (Contributed by Jim Kingdon, 16-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. om )   =>    |- ( ph -> ( G ` suc A ) = ( ( G ` A ) + 1 ) )
Theorem	frec2uzuzd 10636*	The value G (see frec2uz0d 10633) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. om )   =>    |- ( ph -> ( G ` A ) e. ( ZZ>= ` C ) )
Theorem	frec2uzltd 10637*	Less-than relation for G (see frec2uz0d 10633). (Contributed by Jim Kingdon, 16-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. om )   &    |- ( ph -> B e. om )   =>    |- ( ph -> ( A e. B -> ( G ` A ) < ( G ` B ) ) )
Theorem	frec2uzlt2d 10638*	The mapping G (see frec2uz0d 10633) preserves order. (Contributed by Jim Kingdon, 16-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. om )   &    |- ( ph -> B e. om )   =>    |- ( ph -> ( A e. B <-> ( G ` A ) < ( G ` B ) ) )
Theorem	frec2uzrand 10639*	Range of G (see frec2uz0d 10633). (Contributed by Jim Kingdon, 17-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> ran G = ( ZZ>= ` C ) )
Theorem	frec2uzf1od 10640*	G (see frec2uz0d 10633) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> G : om -1-1-onto-> ( ZZ>= ` C ) )
Theorem	frec2uzisod 10641*	G (see frec2uz0d 10633) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> G Isom _E , < ( om , ( ZZ>= ` C ) ) )
Theorem	frecuzrdgrrn 10642*	The function R (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of S. (Contributed by Jim Kingdon, 28-Mar-2022.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   =>    |- ( ( ph /\ D e. om ) -> ( R ` D ) e. ( ( ZZ>= ` C ) X. S ) )
Theorem	frec2uzrdg 10643*	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. This lemma shows that evaluating R at an element of om gives an ordered pair whose first element is the index (translated from om to ( ZZ>= ` C )). See comment in frec2uz0d 10633 which describes G and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> B e. om )   =>    |- ( ph -> ( R ` B ) = <. ( G ` B ) , ( 2nd ` ( R ` B ) ) >. )
Theorem	frecuzrdgrcl 10644*	The function R (used in the definition of the recursive definition generator on upper integers) is a function defined for all natural numbers. (Contributed by Jim Kingdon, 1-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   =>    |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
Theorem	frecuzrdglem 10645*	A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> B e. ( ZZ>= ` C ) )   =>    |- ( ph -> <. B , ( 2nd ` ( R ` ( `' G ` B ) ) ) >. e. ran R )
Theorem	frecuzrdgtcl 10646*	The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10633 for the description of G as the mapping from om to ( ZZ>= ` C ). (Contributed by Jim Kingdon, 26-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> T = ran R )   =>    |- ( ph -> T : ( ZZ>= ` C ) --> S )
Theorem	frecuzrdg0 10647*	Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10633 for the description of G as the mapping from om to ( ZZ>= ` C ). (Contributed by Jim Kingdon, 27-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> T = ran R )   =>    |- ( ph -> ( T ` C ) = A )
Theorem	frecuzrdgsuc 10648*	Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10633 for the description of G as the mapping from om to ( ZZ>= ` C ). (Contributed by Jim Kingdon, 28-May-2020.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. S |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> T = ran R )   =>    |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( T ` ( B + 1 ) ) = ( B F ( T ` B ) ) )
Theorem	frecuzrdgrclt 10649*	The function R (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of S. Similar to frecuzrdgrcl 10644 except that S and T need not be the same. (Contributed by Jim Kingdon, 22-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   =>    |- ( ph -> R : om --> ( ( ZZ>= ` C ) X. S ) )
Theorem	frecuzrdgg 10650*	Lemma for other theorems involving the the recursive definition generator on upper integers. Evaluating R at a natural number gives an ordered pair whose first element is the mapping of that natural number via G. (Contributed by Jim Kingdon, 23-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> N e. om )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> ( 1st ` ( R ` N ) ) = ( G ` N ) )
Theorem	frecuzrdgdomlem 10651*	The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> dom ran R = ( ZZ>= ` C ) )
Theorem	frecuzrdgdom 10652*	The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   =>    |- ( ph -> dom ran R = ( ZZ>= ` C ) )
Theorem	frecuzrdgfunlem 10653*	The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   =>    |- ( ph -> Fun ran R )
Theorem	frecuzrdgfun 10654*	The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   =>    |- ( ph -> Fun ran R )
Theorem	frecuzrdgtclt 10655*	The recursive definition generator on upper integers is a function. (Contributed by Jim Kingdon, 22-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> P = ran R )   =>    |- ( ph -> P : ( ZZ>= ` C ) --> S )
Theorem	frecuzrdg0t 10656*	Initial value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 28-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> P = ran R )   =>    |- ( ph -> ( P ` C ) = A )
Theorem	frecuzrdgsuctlem 10657*	Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10633 for the description of G as the mapping from om to ( ZZ>= ` C ). (Contributed by Jim Kingdon, 29-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> P = ran R )   =>    |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( P ` ( B + 1 ) ) = ( B F ( P ` B ) ) )
Theorem	frecuzrdgsuct 10658*	Successor value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 29-Apr-2022.)
|- ( ph -> C e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. ( ZZ>= ` C ) /\ y e. S ) ) -> ( x F y ) e. S )   &    |- R = frec ( ( x e. ( ZZ>= ` C ) , y e. T |-> <. ( x + 1 ) , ( x F y ) >. ) , <. C , A >. )   &    |- ( ph -> P = ran R )   =>    |- ( ( ph /\ B e. ( ZZ>= ` C ) ) -> ( P ` ( B + 1 ) ) = ( B F ( P ` B ) ) )
Theorem	uzenom 10659	An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> Z ~~ om )
Theorem	frecfzennn 10660	The cardinality of a finite set of sequential integers. (See frec2uz0d 10633 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( N e. NN0 -> ( 1 ... N ) ~~ ( `' G ` N ) )
Theorem	frecfzen2 10661	The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( N e. ( ZZ>= ` M ) -> ( M ... N ) ~~ ( `' G ` ( ( N + 1 ) - M ) ) )
Theorem	frechashgf1o 10662	G maps om one-to-one onto NN0. (Contributed by Jim Kingdon, 19-May-2020.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- G : om -1-1-onto-> NN0
Theorem	frec2uzled 10663*	The mapping G (see frec2uz0d 10633) preserves order. (Contributed by Jim Kingdon, 24-Feb-2022.)
|- ( ph -> C e. ZZ )   &    |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , C )   &    |- ( ph -> A e. om )   &    |- ( ph -> B e. om )   =>    |- ( ph -> ( A C_ B <-> ( G ` A ) <_ ( G ` B ) ) )
Theorem	fzfig 10664	A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M ... N ) e. Fin )
Theorem	fzfigd 10665	Deduction form of fzfig 10664. (Contributed by Jim Kingdon, 21-May-2020.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( M ... N ) e. Fin )
Theorem	fzofig 10666	Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M..^ N ) e. Fin )
Theorem	nn0ennn 10667	The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
|- NN0 ~~ NN
Theorem	nnenom 10668	The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of natural numbers as ordinals). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
|- NN ~~ om
Theorem	nnct 10669	NN is dominated by om. (Contributed by Thierry Arnoux, 29-Dec-2016.)
|- NN ~<_ om
Theorem	uzennn 10670	An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.)
|- ( M e. ZZ -> ( ZZ>= ` M ) ~~ NN )
Theorem	xnn0nnen 10671	The set of extended nonnegative integers is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 14-Jul-2025.)
|- NN0* ~~ NN
Theorem	fnn0nninf 10672*	A function from NN0 into ℕ∞. (Contributed by Jim Kingdon, 16-Jul-2022.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- F = ( n e. om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) )   =>    |- ( F o. `' G ) : NN0 -->ℕ∞
Theorem	fxnn0nninf 10673*	A function from NN0* into ℕ∞. (Contributed by Jim Kingdon, 16-Jul-2022.) TODO: use infnninf 7302 instead of infnninfOLD 7303. More generally, this theorem and most theorems in this section could use an extended G defined by G = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) u. <. om , +oo >. ) and F = ( n e. suc om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) as in nnnninf2 7305.
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- F = ( n e. om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) )   &    |- I = ( ( F o. `' G ) u. { <. +oo , ( om X. { 1o } ) >. } )   =>    |- I :NN0* -->ℕ∞
Theorem	0tonninf 10674*	The mapping of zero into ℕ∞ is the sequence of all zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- F = ( n e. om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) )   &    |- I = ( ( F o. `' G ) u. { <. +oo , ( om X. { 1o } ) >. } )   =>    |- ( I ` 0 ) = ( x e. om |-> (/) )
Theorem	1tonninf 10675*	The mapping of one into ℕ∞ is a sequence which is a one followed by zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- F = ( n e. om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) )   &    |- I = ( ( F o. `' G ) u. { <. +oo , ( om X. { 1o } ) >. } )   =>    |- ( I ` 1 ) = ( x e. om |-> if ( x = (/) , 1o , (/) ) )
Theorem	inftonninf 10676*	The mapping of +oo into ℕ∞ is the sequence of all ones. (Contributed by Jim Kingdon, 17-Jul-2022.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- F = ( n e. om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) )   &    |- I = ( ( F o. `' G ) u. { <. +oo , ( om X. { 1o } ) >. } )   =>    |- ( I ` +oo ) = ( x e. om |-> 1o )
Theorem	nninfinf 10677	ℕ∞ is infinte. (Contributed by Jim Kingdon, 8-Jul-2025.)
|- om ~<_ ℕ∞
4.6.4  Strong induction over upper sets of integers
Theorem	uzsinds 10678*	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 10679*	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 10680*	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 )
4.6.5  The infinite sequence builder "seq"
Syntax	cseq 10681	Extend class notation with recursive sequence builder.
class seq M ( .+ , F )
Definition	df-seqfrec 10682*	Define a general-purpose operation that builds a recursive sequence (i.e., a function on an upper integer set such as NN or NN0) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by seqf 10698, seq3-1 10696 and seq3p1 10699. Typically, those are the main theorems that would be used in practice. The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation +, an input sequence F with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence seq 1 ( + , F ) with values 1, 3/2, 7/4, 15/8,.., so that ( seq 1 ( + , F ) ` 1 ) = 1, ( seq 1 ( + , F ) ` 2 ) = 3/2, etc. In other words, seq M ( + , F ) transforms a sequence F into an infinite series. seq M ( + , F ) ~~> 2 means "the sum of F(n) from n = M to infinity is 2". Since limits are unique (climuni 11819), by climdm 11821 the "sum of F(n) from n = 1 to infinity" can be expressed as ( ~~> ` seq 1 ( + , F ) ) (provided the sequence converges) and evaluates to 2 in this example. Internally, the frec function generates as its values a set of ordered pairs starting at <. M , ( F ` M ) >., with the first member of each pair incremented by one in each successive value. So, the range of frec is exactly the sequence we want, and we just extract the range and throw away the domain. (Contributed by NM, 18-Apr-2005.) (Revised by Jim Kingdon, 4-Nov-2022.)
|- seq M ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
Theorem	seqex 10683	Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
|- seq M ( .+ , F ) e. _V
Theorem	seqeq1 10684	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
|- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )
Theorem	seqeq2 10685	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
|- ( .+ = Q -> seq M ( .+ , F ) = seq M ( Q , F ) )
Theorem	seqeq3 10686	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
|- ( F = G -> seq M ( .+ , F ) = seq M ( .+ , G ) )
Theorem	seqeq1d 10687	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> seq A ( .+ , F ) = seq B ( .+ , F ) )
Theorem	seqeq2d 10688	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> seq M ( A , F ) = seq M ( B , F ) )
Theorem	seqeq3d 10689	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> seq M ( .+ , A ) = seq M ( .+ , B ) )
Theorem	seqeq123d 10690	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
|- ( ph -> M = N )   &    |- ( ph -> .+ = Q )   &    |- ( ph -> F = G )   =>    |- ( ph -> seq M ( .+ , F ) = seq N ( Q , G ) )
Theorem	nfseq 10691	Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x M   &    |- F/_ x .+   &    |- F/_ x F   =>    |- F/_ x seq M ( .+ , F )
Theorem	iseqovex 10692*	Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.)
|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. S ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) e. S )
Theorem	iseqvalcbv 10693*	Changing the bound variables in an expression which appears in some seq related proofs. (Contributed by Jim Kingdon, 28-Apr-2022.)
|- frec ( ( x e. ( ZZ>= ` M ) , y e. T |-> <. ( x + 1 ) , ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( a e. ( ZZ>= ` M ) , b e. T |-> <. ( a + 1 ) , ( a ( c e. ( ZZ>= ` M ) , d e. S |-> ( d .+ ( F ` ( c + 1 ) ) ) ) b ) >. ) , <. M , ( F ` M ) >. )
Theorem	seq3val 10694*	Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10698, seq3-1 10696 and seq3p1 10699, as further development can be done in terms of those. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 4-Nov-2022.)
|- ( ph -> M e. ZZ )   &    |- R = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> seq M ( .+ , F ) = ran R )
Theorem	seqvalcd 10695*	Value of the sequence builder function. Similar to seq3val 10694 but the classes D (type of each term) and C (type of the value we are accumulating) do not need to be the same. (Contributed by Jim Kingdon, 9-Jul-2023.)
|- ( ph -> M e. ZZ )   &    |- R = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( x ( z e. ( ZZ>= ` M ) , w e. C |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. )   &    |- ( ph -> ( F ` M ) e. C )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   =>    |- ( ph -> seq M ( .+ , F ) = ran R )
Theorem	seq3-1 10696*	Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 3-Oct-2022.)
|- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
Theorem	seq1g 10697	Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 19-Aug-2025.)
|- ( ( M e. ZZ /\ F e. V /\ .+ e. W ) -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
Theorem	seqf 10698*	Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. Z ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> seq M ( .+ , F ) : Z --> S )
Theorem	seq3p1 10699*	Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
Theorem	seqp1g 10700	Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 19-Aug-2025.)
|- ( ( N e. ( ZZ>= ` M ) /\ F e. V /\ .+ e. W ) -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )