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Theorem lcomfsup 25916
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 25915 . . 3  |-  ( ph  ->  ( G  o F 
.x.  H ) : I --> B )
11 eldifi 3332 . . . . 5  |-  ( x  e.  ( I  \ 
( `' G "
( _V  \  { Y } ) ) )  ->  x  e.  I
)
12 ffn 5427 . . . . . . . 8  |-  ( G : I --> K  ->  G  Fn  I )
137, 12syl 15 . . . . . . 7  |-  ( ph  ->  G  Fn  I )
1413adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  G  Fn  I )
15 ffn 5427 . . . . . . . 8  |-  ( H : I --> B  ->  H  Fn  I )
168, 15syl 15 . . . . . . 7  |-  ( ph  ->  H  Fn  I )
1716adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  H  Fn  I )
189adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  V )
19 simpr 447 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
20 fnfvof 6132 . . . . . 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 1183 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( G  o F 
.x.  H ) `  x )  =  ( ( G `  x
)  .x.  ( H `  x ) ) )
2211, 21sylan2 460 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( ( G  o F  .x.  H
) `  x )  =  ( ( G `
 x )  .x.  ( H `  x ) ) )
23 ssid 3231 . . . . . . 7  |-  ( `' G " ( _V 
\  { Y }
) )  C_  ( `' G " ( _V 
\  { Y }
) )
2423a1i 10 . . . . . 6  |-  ( ph  ->  ( `' G "
( _V  \  { Y } ) )  C_  ( `' G " ( _V 
\  { Y }
) ) )
257, 24suppssr 5697 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( G `  x )  =  Y )
2625oveq1d 5915 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( ( G `
 x )  .x.  ( H `  x ) )  =  ( Y 
.x.  ( H `  x ) ) )
276adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  W  e.  LMod )
28 ffvelrn 5701 . . . . . . 7  |-  ( ( H : I --> B  /\  x  e.  I )  ->  ( H `  x
)  e.  B )
298, 28sylan 457 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  B )
30 lcomfsup.y . . . . . . 7  |-  Y  =  ( 0g `  F
)
31 lcomfsup.z . . . . . . 7  |-  .0.  =  ( 0g `  W )
325, 2, 4, 30, 31lmod0vs 15712 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( H `  x )  e.  B )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3327, 29, 32syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3411, 33sylan2 460 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3522, 26, 343eqtrd 2352 . . 3  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( ( G  o F  .x.  H
) `  x )  =  .0.  )
3610, 35suppss 5696 . 2  |-  ( ph  ->  ( `' ( G  o F  .x.  H
) " ( _V 
\  {  .0.  }
) )  C_  ( `' G " ( _V 
\  { Y }
) ) )
37 ssfi 7126 . 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 )
381, 36, 37syl2anc 642 1  |-  ( ph  ->  ( `' ( G  o F  .x.  H
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822    \ cdif 3183    C_ wss 3186   {csn 3674   `'ccnv 4725   "cima 4729    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900    o Fcof 6118   Fincfn 6906   Basecbs 13195  Scalarcsca 13258   .scvsca 13259   0gc0g 13449   LModclmod 15676
This theorem is referenced by:  islindf4  26456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-riota 6346  df-er 6702  df-en 6907  df-fin 6910  df-0g 13453  df-mnd 14416  df-grp 14538  df-rng 15389  df-lmod 15678
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