Theorem fsumgcl 11396

Description: Closure for a function used to describe a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)

Hypotheses

Ref	Expression
fsum.1	|- ( k = ( F ` n ) -> B = C )
fsum.2	|- ( ph -> M e. NN )
fsum.3	|- ( ph -> F : ( 1 ... M ) -1-1-onto-> A )
fsum.4	|- ( ( ph /\ k e. A ) -> B e. CC )
fsum.5	|- ( ( ph /\ n e. ( 1 ... M ) ) -> ( G ` n ) = C )

Assertion

Ref	Expression
fsumgcl	|- ( ph -> A. n e. ( 1 ... M ) ( G ` n ) e. CC )

Distinct variable groups:   A, k, n   B, n   C, k   k, F, n   k, G, n   k, M, n   ph, k, n

Allowed substitution hints:   B( k)   C( n)

Proof of Theorem fsumgcl

Step	Hyp	Ref	Expression
1		fsum.5	. . 3 |- ( ( ph /\ n e. ( 1 ... M ) ) -> ( G ` n ) = C )
2		fsum.3	. . . . . . 7 |- ( ph -> F : ( 1 ... M ) -1-1-onto-> A )
3		f1of 5463	. . . . . . 7 |- ( F : ( 1 ... M ) -1-1-onto-> A -> F : ( 1 ... M ) --> A )
4	2, 3	syl 14	. . . . . 6 |- ( ph -> F : ( 1 ... M ) --> A )
5	4	ffvelcdmda 5653	. . . . 5 |- ( ( ph /\ n e. ( 1 ... M ) ) -> ( F ` n ) e. A )
6		fsum.1	. . . . . 6 |- ( k = ( F ` n ) -> B = C )
7	6	adantl 277	. . . . 5 |- ( ( ( ph /\ n e. ( 1 ... M ) ) /\ k = ( F ` n ) ) -> B = C )
8	5, 7	csbied 3105	. . . 4 |- ( ( ph /\ n e. ( 1 ... M ) ) -> [_ ( F ` n ) / k ]_ B = C )
9		fsum.4	. . . . . . 7 |- ( ( ph /\ k e. A ) -> B e. CC )
10	9	ralrimiva 2550	. . . . . 6 |- ( ph -> A. k e. A B e. CC )
11	10	adantr 276	. . . . 5 |- ( ( ph /\ n e. ( 1 ... M ) ) -> A. k e. A B e. CC )
12		nfcsb1v 3092	. . . . . . 7 |- F/_ k [_ ( F ` n ) / k ]_ B
13	12	nfel1 2330	. . . . . 6 |- F/ k [_ ( F ` n ) / k ]_ B e. CC
14		csbeq1a 3068	. . . . . . 7 |- ( k = ( F ` n ) -> B = [_ ( F ` n ) / k ]_ B )
15	14	eleq1d 2246	. . . . . 6 |- ( k = ( F ` n ) -> ( B e. CC <-> [_ ( F ` n ) / k ]_ B e. CC ) )
16	13, 15	rspc 2837	. . . . 5 |- ( ( F ` n ) e. A -> ( A. k e. A B e. CC -> [_ ( F ` n ) / k ]_ B e. CC ) )
17	5, 11, 16	sylc 62	. . . 4 |- ( ( ph /\ n e. ( 1 ... M ) ) -> [_ ( F ` n ) / k ]_ B e. CC )
18	8, 17	eqeltrrd 2255	. . 3 |- ( ( ph /\ n e. ( 1 ... M ) ) -> C e. CC )
19	1, 18	eqeltrd 2254	. 2 |- ( ( ph /\ n e. ( 1 ... M ) ) -> ( G ` n ) e. CC )
20	19	ralrimiva 2550	1 |- ( ph -> A. n e. ( 1 ... M ) ( G ` n ) e. CC )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   [_csb 3059   -->wf 5214   -1-1-onto->wf1o 5217   ` cfv 5218  (class class class)co 5877   CCcc 7811   1c1 7814   NNcn 8921   ...cfz 10010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-f1o 5225  df-fv 5226

This theorem is referenced by:  fsum3  11397  fprodseq  11593