Theorem fsumgcl 12027

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 5592	. . . . . . 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 5790	. . . . 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 3175	. . . 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 2606	. . . . . 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 3161	. . . . . . 7 |- F/_ k [_ ( F ` n ) / k ]_ B
13	12	nfel1 2386	. . . . . 6 |- F/ k [_ ( F ` n ) / k ]_ B e. CC
14		csbeq1a 3137	. . . . . . 7 |- ( k = ( F ` n ) -> B = [_ ( F ` n ) / k ]_ B )
15	14	eleq1d 2300	. . . . . 6 |- ( k = ( F ` n ) -> ( B e. CC <-> [_ ( F ` n ) / k ]_ B e. CC ) )
16	13, 15	rspc 2905	. . . . 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 2309	. . 3 |- ( ( ph /\ n e. ( 1 ... M ) ) -> C e. CC )
19	1, 18	eqeltrd 2308	. 2 |- ( ( ph /\ n e. ( 1 ... M ) ) -> ( G ` n ) e. CC )
20	19	ralrimiva 2606	1 |- ( ph -> A. n e. ( 1 ... M ) ( G ` n ) e. CC )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   [_csb 3128   -->wf 5329   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   CCcc 8090   1c1 8093   NNcn 9202   ...cfz 10305

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-f1o 5340  df-fv 5341

This theorem is referenced by:  fsum3  12028  fprodseq  12224