Theorem fsumgcl 11568

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 5507	. . . . . . 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 5700	. . . . 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 3131	. . . 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 2570	. . . . . 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 3117	. . . . . . 7 |- F/_ k [_ ( F ` n ) / k ]_ B
13	12	nfel1 2350	. . . . . 6 |- F/ k [_ ( F ` n ) / k ]_ B e. CC
14		csbeq1a 3093	. . . . . . 7 |- ( k = ( F ` n ) -> B = [_ ( F ` n ) / k ]_ B )
15	14	eleq1d 2265	. . . . . 6 |- ( k = ( F ` n ) -> ( B e. CC <-> [_ ( F ` n ) / k ]_ B e. CC ) )
16	13, 15	rspc 2862	. . . . 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 2274	. . 3 |- ( ( ph /\ n e. ( 1 ... M ) ) -> C e. CC )
19	1, 18	eqeltrd 2273	. 2 |- ( ( ph /\ n e. ( 1 ... M ) ) -> ( G ` n ) e. CC )
20	19	ralrimiva 2570	1 |- ( ph -> A. n e. ( 1 ... M ) ( G ` n ) e. CC )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   [_csb 3084   -->wf 5255   -1-1-onto->wf1o 5258   ` cfv 5259  (class class class)co 5925   CCcc 7894   1c1 7897   NNcn 9007   ...cfz 10100

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-f1o 5266  df-fv 5267

This theorem is referenced by:  fsum3  11569  fprodseq  11765