Theorem fsumgcl 11897

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 5572	. . . . . . 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 5770	. . . . 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 3171	. . . 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 2603	. . . . . 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 3157	. . . . . . 7 |- F/_ k [_ ( F ` n ) / k ]_ B
13	12	nfel1 2383	. . . . . 6 |- F/ k [_ ( F ` n ) / k ]_ B e. CC
14		csbeq1a 3133	. . . . . . 7 |- ( k = ( F ` n ) -> B = [_ ( F ` n ) / k ]_ B )
15	14	eleq1d 2298	. . . . . 6 |- ( k = ( F ` n ) -> ( B e. CC <-> [_ ( F ` n ) / k ]_ B e. CC ) )
16	13, 15	rspc 2901	. . . . 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 2307	. . 3 |- ( ( ph /\ n e. ( 1 ... M ) ) -> C e. CC )
19	1, 18	eqeltrd 2306	. 2 |- ( ( ph /\ n e. ( 1 ... M ) ) -> ( G ` n ) e. CC )
20	19	ralrimiva 2603	1 |- ( ph -> A. n e. ( 1 ... M ) ( G ` n ) e. CC )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   [_csb 3124   -->wf 5314   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6001   CCcc 7997   1c1 8000   NNcn 9110   ...cfz 10204

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-f1o 5325  df-fv 5326

This theorem is referenced by:  fsum3  11898  fprodseq  12094