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Theorem fsumgcl 11342
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
StepHypRef 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 5440 . . . . . . 7  |-  ( F : ( 1 ... M ) -1-1-onto-> A  ->  F :
( 1 ... M
) --> A )
42, 3syl 14 . . . . . 6  |-  ( ph  ->  F : ( 1 ... M ) --> A )
54ffvelrnda 5629 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... M
) )  ->  ( F `  n )  e.  A )
6 fsum.1 . . . . . 6  |-  ( k  =  ( F `  n )  ->  B  =  C )
76adantl 275 . . . . 5  |-  ( ( ( ph  /\  n  e.  ( 1 ... M
) )  /\  k  =  ( F `  n ) )  ->  B  =  C )
85, 7csbied 3095 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... M
) )  ->  [_ ( F `  n )  /  k ]_ B  =  C )
9 fsum.4 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
109ralrimiva 2543 . . . . . 6  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
1110adantr 274 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... M
) )  ->  A. k  e.  A  B  e.  CC )
12 nfcsb1v 3082 . . . . . . 7  |-  F/_ k [_ ( F `  n
)  /  k ]_ B
1312nfel1 2323 . . . . . 6  |-  F/ k
[_ ( F `  n )  /  k ]_ B  e.  CC
14 csbeq1a 3058 . . . . . . 7  |-  ( k  =  ( F `  n )  ->  B  =  [_ ( F `  n )  /  k ]_ B )
1514eleq1d 2239 . . . . . 6  |-  ( k  =  ( F `  n )  ->  ( B  e.  CC  <->  [_ ( F `
 n )  / 
k ]_ B  e.  CC ) )
1613, 15rspc 2828 . . . . 5  |-  ( ( F `  n )  e.  A  ->  ( A. k  e.  A  B  e.  CC  ->  [_ ( F `  n
)  /  k ]_ B  e.  CC )
)
175, 11, 16sylc 62 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... M
) )  ->  [_ ( F `  n )  /  k ]_ B  e.  CC )
188, 17eqeltrrd 2248 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... M
) )  ->  C  e.  CC )
191, 18eqeltrd 2247 . 2  |-  ( (
ph  /\  n  e.  ( 1 ... M
) )  ->  ( G `  n )  e.  CC )
2019ralrimiva 2543 1  |-  ( ph  ->  A. n  e.  ( 1 ... M ) ( G `  n
)  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   A.wral 2448   [_csb 3049   -->wf 5192   -1-1-onto->wf1o 5195   ` cfv 5196  (class class class)co 5851   CCcc 7765   1c1 7768   NNcn 8871   ...cfz 9958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-f1o 5203  df-fv 5204
This theorem is referenced by:  fsum3  11343  fprodseq  11539
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