P. 104 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10301-10400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	modqsubdir 10301	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 10302	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 10303*	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 10304	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 10305	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 10306	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 10307*	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 10308*	The value of G (see frec2uz0d 10307) 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 10309*	The value of G (see frec2uz0d 10307) 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 10310*	The value G (see frec2uz0d 10307) 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 10311*	Less-than relation for G (see frec2uz0d 10307). (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 10312*	The mapping G (see frec2uz0d 10307) 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 10313*	Range of G (see frec2uz0d 10307). (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 10314*	G (see frec2uz0d 10307) 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 10315*	G (see frec2uz0d 10307) 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 10316*	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 10317*	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 10307 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 10318*	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 10319*	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 10320*	The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10307 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 10321*	Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10307 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 10322*	Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10307 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 10323*	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 10318 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 10324*	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 10325*	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 10326*	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 10327*	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 10328*	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 10329*	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 10330*	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 10331*	Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10307 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 10332*	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 10333	An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> Z ~~ om )
Theorem	frecfzennn 10334	The cardinality of a finite set of sequential integers. (See frec2uz0d 10307 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 10335	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 10336	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 10337*	The mapping G (see frec2uz0d 10307) 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 10338	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 10339	Deduction form of fzfig 10338. (Contributed by Jim Kingdon, 21-May-2020.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( M ... N ) e. Fin )
Theorem	fzofig 10340	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 10341	The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
|- NN0 ~~ NN
Theorem	nnenom 10342	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 10343	NN is dominated by om. (Contributed by Thierry Arnoux, 29-Dec-2016.)
|- NN ~<_ om
Theorem	uzennn 10344	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	fnn0nninf 10345*	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 10346*	A function from NN0* into ℕ∞. (Contributed by Jim Kingdon, 16-Jul-2022.) TODO: use infnninf 7069 instead of infnninfOLD 7070. 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 7072.
|- 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 10347*	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 10348*	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 10349*	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 )
4.6.4  Strong induction over upper sets of integers
Theorem	uzsinds 10350*	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 10351*	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 10352*	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 10353	Extend class notation with recursive sequence builder.
class seq M ( .+ , F )
Definition	df-seqfrec 10354*	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 10369, seq3-1 10368 and seq3p1 10370. 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 11201), by climdm 11203 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 10355	Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
|- seq M ( .+ , F ) e. _V
Theorem	seqeq1 10356	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
|- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )
Theorem	seqeq2 10357	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
|- ( .+ = Q -> seq M ( .+ , F ) = seq M ( Q , F ) )
Theorem	seqeq3 10358	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
|- ( F = G -> seq M ( .+ , F ) = seq M ( .+ , G ) )
Theorem	seqeq1d 10359	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 10360	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 10361	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 10362	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 10363	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 10364*	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 10365*	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 10366*	Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10369, seq3-1 10368 and seq3p1 10370, 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 10367*	Value of the sequence builder function. Similar to seq3val 10366 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 10368*	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	seqf 10369*	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 10370*	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	seqovcd 10371*	A closure law for the recursive sequence builder. This is a lemma for theorems such as seqf2 10372 and seq1cd 10373 and is unlikely to be needed once such theorems are proved. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   =>    |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. C ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. C |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) e. C )
Theorem	seqf2 10372*	Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 7-Jul-2023.)
|- ( ph -> ( F ` M ) e. C )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   =>    |- ( ph -> seq M ( .+ , F ) : Z --> C )
Theorem	seq1cd 10373*	Initial value of the recursive sequence builder. A version of seq3-1 10368 which provides two classes D and C for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> ( F ` M ) e. C )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   =>    |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
Theorem	seqp1cd 10374*	Value of the sequence builder function at a successor. A version of seq3p1 10370 which provides two classes D and C for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ph -> N e. ( ZZ>= ` 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 ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
Theorem	seq3clss 10375*	Closure property of the recursive sequence builder. (Contributed by Jim Kingdon, 28-Sep-2022.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. T )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ ( x e. T /\ y e. T ) ) -> ( x .+ y ) e. T )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) e. S )
Theorem	seq3m1 10376*	Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 3-Nov-2022.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) )   &    |- ( ( 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 ) = ( ( seq M ( .+ , F ) ` ( N - 1 ) ) .+ ( F ` N ) ) )
Theorem	seq3fveq2 10377*	Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
|- ( ph -> K e. ( ZZ>= ` M ) )   &    |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ph -> N e. ( ZZ>= ` K ) )   &    |- ( ( ph /\ k e. ( ( K + 1 ) ... N ) ) -> ( F ` k ) = ( G ` k ) )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , G ) ` N ) )
Theorem	seq3feq2 10378*	Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
|- ( ph -> K e. ( ZZ>= ` M ) )   &    |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( G ` K ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` ( K + 1 ) ) ) -> ( F ` k ) = ( G ` k ) )   =>    |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` K ) ) = seq K ( .+ , G ) )
Theorem	seq3fveq 10379*	Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = ( G ` k ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq M ( .+ , G ) ` N ) )
Theorem	seq3feq 10380*	Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
|- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( G ` k ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> seq M ( .+ , F ) = seq M ( .+ , G ) )
Theorem	seq3shft2 10381*	Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> K e. ZZ )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = ( G ` ( k + K ) ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + K ) ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq ( M + K ) ( .+ , G ) ` ( N + K ) ) )
Theorem	serf 10382*	An infinite series of complex terms is a function from NN to CC. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   =>    |- ( ph -> seq M ( + , F ) : Z --> CC )
Theorem	serfre 10383*	An infinite series of real numbers is a function from NN to RR. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   =>    |- ( ph -> seq M ( + , F ) : Z --> RR )
Theorem	monoord 10384*	Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )   =>    |- ( ph -> ( F ` M ) <_ ( F ` N ) )
Theorem	monoord2 10385*	Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> ( F ` ( k + 1 ) ) <_ ( F ` k ) )   =>    |- ( ph -> ( F ` N ) <_ ( F ` M ) )
Theorem	ser3mono 10386*	The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
|- ( ph -> K e. ( ZZ>= ` M ) )   &    |- ( ph -> N e. ( ZZ>= ` K ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR )   &    |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> 0 <_ ( F ` x ) )   =>    |- ( ph -> ( seq M ( + , F ) ` K ) <_ ( seq M ( + , F ) ` N ) )
Theorem	seq3split 10387*	Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) (Revised by Jim Kingdon, 21-Oct-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) )   &    |- ( ph -> M e. ( ZZ>= ` K ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( F ` x ) e. S )   =>    |- ( ph -> ( seq K ( .+ , F ) ` N ) = ( ( seq K ( .+ , F ) ` M ) .+ ( seq ( M + 1 ) ( .+ , F ) ` N ) ) )
Theorem	seq3-1p 10388*	Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( ( F ` M ) .+ ( seq ( M + 1 ) ( .+ , F ) ` N ) ) )
Theorem	seq3caopr3 10389*	Lemma for seq3caopr2 10390. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) )   &    |- ( ( ph /\ n e. ( M..^ N ) ) -> ( ( ( seq M ( .+ , F ) ` n ) Q ( seq M ( .+ , G ) ` n ) ) .+ ( ( F ` ( n + 1 ) ) Q ( G ` ( n + 1 ) ) ) ) = ( ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) Q ( ( seq M ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) Q ( seq M ( .+ , G ) ` N ) ) )
Theorem	seq3caopr2 10390*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   &    |- ( ( ph /\ ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ) -> ( ( x Q z ) .+ ( y Q w ) ) = ( ( x .+ y ) Q ( z .+ w ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) Q ( seq M ( .+ , G ) ` N ) ) )
Theorem	seq3caopr 10391*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) .+ ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) .+ ( seq M ( .+ , G ) ` N ) ) )
Theorem	iseqf1olemkle 10392*	Lemma for seq3f1o 10412. (Contributed by Jim Kingdon, 21-Aug-2022.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   =>    |- ( ph -> K <_ ( `' J ` K ) )
Theorem	iseqf1olemklt 10393*	Lemma for seq3f1o 10412. (Contributed by Jim Kingdon, 21-Aug-2022.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )   &    |- ( ph -> K =/= ( `' J ` K ) )   =>    |- ( ph -> K < ( `' J ` K ) )
Theorem	iseqf1olemqcl 10394	Lemma for seq3f1o 10412. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   =>    |- ( ph -> if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) e. ( M ... N ) )
Theorem	iseqf1olemqval 10395*	Lemma for seq3f1o 10412. Value of the function Q. (Contributed by Jim Kingdon, 28-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   =>    |- ( ph -> ( Q ` A ) = if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) )
Theorem	iseqf1olemnab 10396*	Lemma for seq3f1o 10412. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ph -> B e. ( M ... N ) )   &    |- ( ph -> ( Q ` A ) = ( Q ` B ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   =>    |- ( ph -> -. ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) )
Theorem	iseqf1olemab 10397*	Lemma for seq3f1o 10412. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ph -> B e. ( M ... N ) )   &    |- ( ph -> ( Q ` A ) = ( Q ` B ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ph -> A e. ( K ... ( `' J ` K ) ) )   &    |- ( ph -> B e. ( K ... ( `' J ` K ) ) )   =>    |- ( ph -> A = B )
Theorem	iseqf1olemnanb 10398*	Lemma for seq3f1o 10412. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ph -> B e. ( M ... N ) )   &    |- ( ph -> ( Q ` A ) = ( Q ` B ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ph -> -. A e. ( K ... ( `' J ` K ) ) )   &    |- ( ph -> -. B e. ( K ... ( `' J ` K ) ) )   =>    |- ( ph -> A = B )
Theorem	iseqf1olemqf 10399*	Lemma for seq3f1o 10412. Domain and codomain of Q. (Contributed by Jim Kingdon, 26-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   =>    |- ( ph -> Q : ( M ... N ) --> ( M ... N ) )
Theorem	iseqf1olemmo 10400*	Lemma for seq3f1o 10412. Showing that Q is one-to-one. (Contributed by Jim Kingdon, 27-Aug-2022.)
|- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )   &    |- ( ph -> A e. ( M ... N ) )   &    |- ( ph -> B e. ( M ... N ) )   &    |- ( ph -> ( Q ` A ) = ( Q ` B ) )   =>    |- ( ph -> A = B )