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Theorem lcomfsupOLD 16983
Description: A linear-combination sum is finitely supported if the coefficients are. (Contributed by Stefan O'Rear, 28-Feb-2015.) Obsolete version of lcomfsupp 16984 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 16982 . . 3  |-  ( ph  ->  ( G  oF  .x.  H ) : I --> B )
11 eldifi 3477 . . . . 5  |-  ( x  e.  ( I  \ 
( `' G "
( _V  \  { Y } ) ) )  ->  x  e.  I
)
12 ffn 5558 . . . . . . . 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 5558 . . . . . . . 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 6332 . . . . . 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 1219 . . . . 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 3374 . . . . . . 7  |-  ( `' G " ( _V 
\  { Y }
) )  C_  ( `' G " ( _V 
\  { Y }
) )
2423a1i 11 . . . . . 6  |-  ( ph  ->  ( `' G "
( _V  \  { Y } ) )  C_  ( `' G " ( _V 
\  { Y }
) ) )
257, 24suppssrOLD 5836 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( G `  x )  =  Y )
2625oveq1d 6105 . . . 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 5842 . . . . . 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 16980 . . . . . 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 2478 . . 3  |-  ( (
ph  /\  x  e.  ( I  \  ( `' G " ( _V 
\  { Y }
) ) ) )  ->  ( ( G  oF  .x.  H
) `  x )  =  .0.  )
3510, 34suppssOLD 5835 . 2  |-  ( ph  ->  ( `' ( G  oF  .x.  H
) " ( _V 
\  {  .0.  }
) )  C_  ( `' G " ( _V 
\  { Y }
) ) )
36 ssfi 7532 . 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 1369    e. wcel 1756   _Vcvv 2971    \ cdif 3324    C_ wss 3327   {csn 3876   `'ccnv 4838   "cima 4842    Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6090    oFcof 6317   Fincfn 7309   Basecbs 14173  Scalarcsca 14240   .scvsca 14241   0gc0g 14377   LModclmod 16947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-er 7100  df-en 7310  df-fin 7313  df-0g 14379  df-mnd 15414  df-grp 15544  df-rng 16646  df-lmod 16949
This theorem is referenced by: (None)
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