Theorem etransclem43

Description: G is a continuous function. (Contributed by Glauco Siliprandi, 5-Apr-2020)

	Ref	Expression
Hypotheses	etransclem43.s	|- ( ph -> S e. { RR , CC } )
	etransclem43.x	|- ( ph -> X e. ( ( TopOpen ` CCfld ) |`t S ) )
	etransclem43.p	|- ( ph -> P e. NN )
	etransclem43.m	|- ( ph -> M e. NN0 )
	etransclem43.f	|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
	etransclem43.g	|- G = ( x e. X |-> sum_ i e. ( 0 ... R ) ( ( ( S Dn F ) ` i ) ` x ) )
Assertion	etransclem43	|- ( ph -> G e. ( X -cn-> CC ) )

Proof

Step	Hyp	Ref	Expression
1		etransclem43.s	|- ( ph -> S e. { RR , CC } )
2		etransclem43.x	|- ( ph -> X e. ( ( TopOpen ` CCfld ) |`t S ) )
3		etransclem43.p	|- ( ph -> P e. NN )
4		etransclem43.m	|- ( ph -> M e. NN0 )
5		etransclem43.f	|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
6		etransclem43.g	|- G = ( x e. X |-> sum_ i e. ( 0 ... R ) ( ( ( S Dn F ) ` i ) ` x ) )
7	1 2	dvdmsscn	|- ( ph -> X C_ CC )
8		fzfid	|- ( ph -> ( 0 ... R ) e. Fin )
9	1	adantr	|- ( ( ph /\ i e. ( 0 ... R ) ) -> S e. { RR , CC } )
10	2	adantr	|- ( ( ph /\ i e. ( 0 ... R ) ) -> X e. ( ( TopOpen ` CCfld ) |`t S ) )
11	3	adantr	|- ( ( ph /\ i e. ( 0 ... R ) ) -> P e. NN )
12	4	adantr	|- ( ( ph /\ i e. ( 0 ... R ) ) -> M e. NN0 )
13		elfznn0	|- ( i e. ( 0 ... R ) -> i e. NN0 )
14	13	adantl	|- ( ( ph /\ i e. ( 0 ... R ) ) -> i e. NN0 )
15	9 10 11 12 5 14	etransclem33	|- ( ( ph /\ i e. ( 0 ... R ) ) -> ( ( S Dn F ) ` i ) : X --> CC )
16	15	feqmptd	|- ( ( ph /\ i e. ( 0 ... R ) ) -> ( ( S Dn F ) ` i ) = ( x e. X |-> ( ( ( S Dn F ) ` i ) ` x ) ) )
17	9 10 11 12 5 14	etransclem40	|- ( ( ph /\ i e. ( 0 ... R ) ) -> ( ( S Dn F ) ` i ) e. ( X -cn-> CC ) )
18	16 17	eqeltrrd	|- ( ( ph /\ i e. ( 0 ... R ) ) -> ( x e. X |-> ( ( ( S Dn F ) ` i ) ` x ) ) e. ( X -cn-> CC ) )
19	7 8 18	fsumcncf	|- ( ph -> ( x e. X |-> sum_ i e. ( 0 ... R ) ( ( ( S Dn F ) ` i ) ` x ) ) e. ( X -cn-> CC ) )
20	6 19	eqeltrid	|- ( ph -> G e. ( X -cn-> CC ) )