Theorem fsumgcl 11697

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 5522	. . . . . . 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 5715	. . . . 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 3140	. . . 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 2579	. . . . . 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 3126	. . . . . . 7 |- F/_ k [_ ( F ` n ) / k ]_ B
13	12	nfel1 2359	. . . . . 6 |- F/ k [_ ( F ` n ) / k ]_ B e. CC
14		csbeq1a 3102	. . . . . . 7 |- ( k = ( F ` n ) -> B = [_ ( F ` n ) / k ]_ B )
15	14	eleq1d 2274	. . . . . 6 |- ( k = ( F ` n ) -> ( B e. CC <-> [_ ( F ` n ) / k ]_ B e. CC ) )
16	13, 15	rspc 2871	. . . . 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 2283	. . 3 |- ( ( ph /\ n e. ( 1 ... M ) ) -> C e. CC )
19	1, 18	eqeltrd 2282	. 2 |- ( ( ph /\ n e. ( 1 ... M ) ) -> ( G ` n ) e. CC )
20	19	ralrimiva 2579	1 |- ( ph -> A. n e. ( 1 ... M ) ( G ` n ) e. CC )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   A.wral 2484   [_csb 3093   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5944   CCcc 7923   1c1 7926   NNcn 9036   ...cfz 10130

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-f1o 5278  df-fv 5279

This theorem is referenced by:  fsum3  11698  fprodseq  11894