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Theorem lcomfsup 26747
Description: A linear-combination sum is finitely supported if the coefficients are. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Hypotheses
Ref Expression
lcomf.f  |-  F  =  (Scalar `  W )
lcomf.k  |-  K  =  ( Base `  F
)
lcomf.s  |-  .x.  =  ( .s `  W )
lcomf.b  |-  B  =  ( Base `  W
)
lcomf.w  |-  ( ph  ->  W  e.  LMod )
lcomf.g  |-  ( ph  ->  G : I --> K )
lcomf.h  |-  ( ph  ->  H : I --> B )
lcomf.i  |-  ( ph  ->  I  e.  V )
lcomfsup.z  |-  .0.  =  ( 0g `  W )
lcomfsup.y  |-  Y  =  ( 0g `  F
)
lcomfsup.j  |-  ( ph  ->  ( `' G "
( _V  \  { Y } ) )  e. 
Fin )
Assertion
Ref Expression
lcomfsup  |-  ( ph  ->  ( `' ( G  o F  .x.  H
) " ( _V 
\  {  .0.  }
) )  e.  Fin )

Proof of Theorem lcomfsup
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lcomfsup.j . 2  |-  ( ph  ->  ( `' G "
( _V  \  { Y } ) )  e. 
Fin )
2 lcomf.f . . . 4  |-  F  =  (Scalar `  W )
3 lcomf.k . . . 4  |-  K  =  ( Base `  F
)
4 lcomf.s . . . 4  |-  .x.  =  ( .s `  W )
5 lcomf.b . . . 4  |-  B  =  ( Base `  W
)
6 lcomf.w . . . 4  |-  ( ph  ->  W  e.  LMod )
7 lcomf.g . . . 4  |-  ( ph  ->  G : I --> K )
8 lcomf.h . . . 4  |-  ( ph  ->  H : I --> B )
9 lcomf.i . . . 4  |-  ( ph  ->  I  e.  V )
102, 3, 4, 5, 6, 7, 8, 9lcomf 26746 . . 3  |-  ( ph  ->  ( G  o F 
.x.  H ) : I --> B )
11 eldifi 3469 . . . . 5  |-  ( x  e.  ( I  \ 
( `' G "
( _V  \  { Y } ) ) )  ->  x  e.  I
)
12 ffn 5591 . . . . . . . 8  |-  ( G : I --> K  ->  G  Fn  I )
137, 12syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  I )
1413adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  G  Fn  I )
15 ffn 5591 . . . . . . . 8  |-  ( H : I --> B  ->  H  Fn  I )
168, 15syl 16 . . . . . . 7  |-  ( ph  ->  H  Fn  I )
1716adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  H  Fn  I )
189adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  V )
19 simpr 448 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
20 fnfvof 6317 . . . . . 6  |-  ( ( ( G  Fn  I  /\  H  Fn  I
)  /\  ( I  e.  V  /\  x  e.  I ) )  -> 
( ( G  o F  .x.  H ) `  x )  =  ( ( G `  x
)  .x.  ( H `  x ) ) )
2114, 17, 18, 19, 20syl22anc 1185 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( G  o F 
.x.  H ) `  x )  =  ( ( G `  x
)  .x.  ( H `  x ) ) )
2211, 21sylan2 461 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( ( G  o F  .x.  H
) `  x )  =  ( ( G `
 x )  .x.  ( H `  x ) ) )
23 ssid 3367 . . . . . . 7  |-  ( `' G " ( _V 
\  { Y }
) )  C_  ( `' G " ( _V 
\  { Y }
) )
2423a1i 11 . . . . . 6  |-  ( ph  ->  ( `' G "
( _V  \  { Y } ) )  C_  ( `' G " ( _V 
\  { Y }
) ) )
257, 24suppssr 5864 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( G `  x )  =  Y )
2625oveq1d 6096 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( ( G `
 x )  .x.  ( H `  x ) )  =  ( Y 
.x.  ( H `  x ) ) )
276adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  W  e.  LMod )
288ffvelrnda 5870 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  B )
29 lcomfsup.y . . . . . . 7  |-  Y  =  ( 0g `  F
)
30 lcomfsup.z . . . . . . 7  |-  .0.  =  ( 0g `  W )
315, 2, 4, 29, 30lmod0vs 15983 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( H `  x )  e.  B )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3227, 28, 31syl2anc 643 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3311, 32sylan2 461 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3422, 26, 333eqtrd 2472 . . 3  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( ( G  o F  .x.  H
) `  x )  =  .0.  )
3510, 34suppss 5863 . 2  |-  ( ph  ->  ( `' ( G  o F  .x.  H
) " ( _V 
\  {  .0.  }
) )  C_  ( `' G " ( _V 
\  { Y }
) ) )
36 ssfi 7329 . 2  |-  ( ( ( `' G "
( _V  \  { Y } ) )  e. 
Fin  /\  ( `' ( G  o F  .x.  H ) " ( _V  \  {  .0.  }
) )  C_  ( `' G " ( _V 
\  { Y }
) ) )  -> 
( `' ( G  o F  .x.  H
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
371, 35, 36syl2anc 643 1  |-  ( ph  ->  ( `' ( G  o F  .x.  H
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    C_ wss 3320   {csn 3814   `'ccnv 4877   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   Fincfn 7109   Basecbs 13469  Scalarcsca 13532   .scvsca 13533   0gc0g 13723   LModclmod 15950
This theorem is referenced by:  islindf4  27285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-riota 6549  df-er 6905  df-en 7110  df-fin 7113  df-0g 13727  df-mnd 14690  df-grp 14812  df-rng 15663  df-lmod 15952
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