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Theorem lcomfsupOLD 17397
Description: A linear-combination sum is finitely supported if the coefficients are. (Contributed by Stefan O'Rear, 28-Feb-2015.) Obsolete version of lcomfsupp 17398 as of 15-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
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 )
lcomfsupp.z  |-  .0.  =  ( 0g `  W )
lcomfsupp.y  |-  Y  =  ( 0g `  F
)
lcomfsupOLD.j  |-  ( ph  ->  ( `' G "
( _V  \  { Y } ) )  e. 
Fin )
Assertion
Ref Expression
lcomfsupOLD  |-  ( ph  ->  ( `' ( G  oF  .x.  H
) " ( _V 
\  {  .0.  }
) )  e.  Fin )

Proof of Theorem lcomfsupOLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lcomfsupOLD.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 17396 . . 3  |-  ( ph  ->  ( G  oF  .x.  H ) : I --> B )
11 eldifi 3631 . . . . 5  |-  ( x  e.  ( I  \ 
( `' G "
( _V  \  { Y } ) ) )  ->  x  e.  I
)
12 ffn 5736 . . . . . . . 8  |-  ( G : I --> K  ->  G  Fn  I )
137, 12syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  I )
1413adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  G  Fn  I )
15 ffn 5736 . . . . . . . 8  |-  ( H : I --> B  ->  H  Fn  I )
168, 15syl 16 . . . . . . 7  |-  ( ph  ->  H  Fn  I )
1716adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  H  Fn  I )
189adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  V )
19 simpr 461 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
20 fnfvof 6547 . . . . . 6  |-  ( ( ( G  Fn  I  /\  H  Fn  I
)  /\  ( I  e.  V  /\  x  e.  I ) )  -> 
( ( G  oF  .x.  H ) `  x )  =  ( ( G `  x
)  .x.  ( H `  x ) ) )
2114, 17, 18, 19, 20syl22anc 1229 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( G  oF  .x.  H ) `  x )  =  ( ( G `  x
)  .x.  ( H `  x ) ) )
2211, 21sylan2 474 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( ( G  oF  .x.  H
) `  x )  =  ( ( G `
 x )  .x.  ( H `  x ) ) )
23 ssid 3528 . . . . . . 7  |-  ( `' G " ( _V 
\  { Y }
) )  C_  ( `' G " ( _V 
\  { Y }
) )
2423a1i 11 . . . . . 6  |-  ( ph  ->  ( `' G "
( _V  \  { Y } ) )  C_  ( `' G " ( _V 
\  { Y }
) ) )
257, 24suppssrOLD 6021 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( G `  x )  =  Y )
2625oveq1d 6309 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( ( G `
 x )  .x.  ( H `  x ) )  =  ( Y 
.x.  ( H `  x ) ) )
276adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  W  e.  LMod )
288ffvelrnda 6031 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  B )
29 lcomfsupp.y . . . . . . 7  |-  Y  =  ( 0g `  F
)
30 lcomfsupp.z . . . . . . 7  |-  .0.  =  ( 0g `  W )
315, 2, 4, 29, 30lmod0vs 17393 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( H `  x )  e.  B )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3227, 28, 31syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3311, 32sylan2 474 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3422, 26, 333eqtrd 2512 . . 3  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( ( G  oF  .x.  H
) `  x )  =  .0.  )
3510, 34suppssOLD 6020 . 2  |-  ( ph  ->  ( `' ( G  oF  .x.  H
) " ( _V 
\  {  .0.  }
) )  C_  ( `' G " ( _V 
\  { Y }
) ) )
36 ssfi 7750 . 2  |-  ( ( ( `' G "
( _V  \  { Y } ) )  e. 
Fin  /\  ( `' ( G  oF  .x.  H ) " ( _V  \  {  .0.  }
) )  C_  ( `' G " ( _V 
\  { Y }
) ) )  -> 
( `' ( G  oF  .x.  H
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
371, 35, 36syl2anc 661 1  |-  ( ph  ->  ( `' ( G  oF  .x.  H
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    \ cdif 3478    C_ wss 3481   {csn 4032   `'ccnv 5003   "cima 5007    Fn wfn 5588   -->wf 5589   ` cfv 5593  (class class class)co 6294    oFcof 6532   Fincfn 7526   Basecbs 14502  Scalarcsca 14570   .scvsca 14571   0gc0g 14707   LModclmod 17360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-om 6695  df-er 7321  df-en 7527  df-fin 7530  df-0g 14709  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-grp 15906  df-ring 17049  df-lmod 17362
This theorem is referenced by: (None)
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