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Theorem lcomfsupOLD 17423
Description: A linear-combination sum is finitely supported if the coefficients are. (Contributed by Stefan O'Rear, 28-Feb-2015.) Obsolete version of lcomfsupp 17424 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 17422 . . 3  |-  ( ph  ->  ( G  oF  .x.  H ) : I --> B )
11 eldifi 3611 . . . . 5  |-  ( x  e.  ( I  \ 
( `' G "
( _V  \  { Y } ) ) )  ->  x  e.  I
)
12 ffn 5721 . . . . . . . 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 5721 . . . . . . . 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 6538 . . . . . 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 1230 . . . . 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 3508 . . . . . . 7  |-  ( `' G " ( _V 
\  { Y }
) )  C_  ( `' G " ( _V 
\  { Y }
) )
2423a1i 11 . . . . . 6  |-  ( ph  ->  ( `' G "
( _V  \  { Y } ) )  C_  ( `' G " ( _V 
\  { Y }
) ) )
257, 24suppssrOLD 6006 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( G `  x )  =  Y )
2625oveq1d 6296 . . . 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 6016 . . . . . 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 17419 . . . . . 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 2488 . . 3  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( ( G  oF  .x.  H
) `  x )  =  .0.  )
3510, 34suppssOLD 6005 . 2  |-  ( ph  ->  ( `' ( G  oF  .x.  H
) " ( _V 
\  {  .0.  }
) )  C_  ( `' G " ( _V 
\  { Y }
) ) )
36 ssfi 7742 . 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 1383    e. wcel 1804   _Vcvv 3095    \ cdif 3458    C_ wss 3461   {csn 4014   `'ccnv 4988   "cima 4992    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523   Fincfn 7518   Basecbs 14509  Scalarcsca 14577   .scvsca 14578   0gc0g 14714   LModclmod 17386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-er 7313  df-en 7519  df-fin 7522  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-ring 17074  df-lmod 17388
This theorem is referenced by: (None)
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