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Theorem fsumgcl 10764
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 5247 . . . . . . 7  |-  ( F : ( 1 ... M ) -1-1-onto-> A  ->  F :
( 1 ... M
) --> A )
42, 3syl 14 . . . . . 6  |-  ( ph  ->  F : ( 1 ... M ) --> A )
54ffvelrnda 5428 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... M
) )  ->  ( F `  n )  e.  A )
6 fsum.1 . . . . . 6  |-  ( k  =  ( F `  n )  ->  B  =  C )
76adantl 271 . . . . 5  |-  ( ( ( ph  /\  n  e.  ( 1 ... M
) )  /\  k  =  ( F `  n ) )  ->  B  =  C )
85, 7csbied 2974 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... M
) )  ->  [_ ( F `  n )  /  k ]_ B  =  C )
9 fsum.4 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
109ralrimiva 2446 . . . . . 6  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
1110adantr 270 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... M
) )  ->  A. k  e.  A  B  e.  CC )
12 nfcsb1v 2963 . . . . . . 7  |-  F/_ k [_ ( F `  n
)  /  k ]_ B
1312nfel1 2239 . . . . . 6  |-  F/ k
[_ ( F `  n )  /  k ]_ B  e.  CC
14 csbeq1a 2941 . . . . . . 7  |-  ( k  =  ( F `  n )  ->  B  =  [_ ( F `  n )  /  k ]_ B )
1514eleq1d 2156 . . . . . 6  |-  ( k  =  ( F `  n )  ->  ( B  e.  CC  <->  [_ ( F `
 n )  / 
k ]_ B  e.  CC ) )
1613, 15rspc 2716 . . . . 5  |-  ( ( F `  n )  e.  A  ->  ( A. k  e.  A  B  e.  CC  ->  [_ ( F `  n
)  /  k ]_ B  e.  CC )
)
175, 11, 16sylc 61 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... M
) )  ->  [_ ( F `  n )  /  k ]_ B  e.  CC )
188, 17eqeltrrd 2165 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... M
) )  ->  C  e.  CC )
191, 18eqeltrd 2164 . 2  |-  ( (
ph  /\  n  e.  ( 1 ... M
) )  ->  ( G `  n )  e.  CC )
2019ralrimiva 2446 1  |-  ( ph  ->  A. n  e.  ( 1 ... M ) ( G `  n
)  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   A.wral 2359   [_csb 2933   -->wf 5006   -1-1-onto->wf1o 5009   ` cfv 5010  (class class class)co 5644   CCcc 7338   1c1 7341   NNcn 8412   ...cfz 9414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-f1o 5017  df-fv 5018
This theorem is referenced by:  fisum  10765  fsum3  10766
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