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Theorem lcomfsupp 18063
Description: A linear-combination sum is finitely supported if the coefficients are. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by AV, 15-Jul-2019.)
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
)
lcomfsupp.j  |-  ( ph  ->  G finSupp  Y )
Assertion
Ref Expression
lcomfsupp  |-  ( ph  ->  ( G  oF  .x.  H ) finSupp  .0.  )

Proof of Theorem lcomfsupp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lcomfsupp.j . . . 4  |-  ( ph  ->  G finSupp  Y )
21fsuppimpd 7896 . . 3  |-  ( ph  ->  ( G supp  Y )  e.  Fin )
3 lcomf.f . . . . 5  |-  F  =  (Scalar `  W )
4 lcomf.k . . . . 5  |-  K  =  ( Base `  F
)
5 lcomf.s . . . . 5  |-  .x.  =  ( .s `  W )
6 lcomf.b . . . . 5  |-  B  =  ( Base `  W
)
7 lcomf.w . . . . 5  |-  ( ph  ->  W  e.  LMod )
8 lcomf.g . . . . 5  |-  ( ph  ->  G : I --> K )
9 lcomf.h . . . . 5  |-  ( ph  ->  H : I --> B )
10 lcomf.i . . . . 5  |-  ( ph  ->  I  e.  V )
113, 4, 5, 6, 7, 8, 9, 10lcomf 18062 . . . 4  |-  ( ph  ->  ( G  oF  .x.  H ) : I --> B )
12 eldifi 3593 . . . . . 6  |-  ( x  e.  ( I  \ 
( G supp  Y )
)  ->  x  e.  I )
13 ffn 5746 . . . . . . . . 9  |-  ( G : I --> K  ->  G  Fn  I )
148, 13syl 17 . . . . . . . 8  |-  ( ph  ->  G  Fn  I )
1514adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  G  Fn  I )
16 ffn 5746 . . . . . . . . 9  |-  ( H : I --> B  ->  H  Fn  I )
179, 16syl 17 . . . . . . . 8  |-  ( ph  ->  H  Fn  I )
1817adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  H  Fn  I )
1910adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  V )
20 simpr 462 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
21 fnfvof 6559 . . . . . . 7  |-  ( ( ( G  Fn  I  /\  H  Fn  I
)  /\  ( I  e.  V  /\  x  e.  I ) )  -> 
( ( G  oF  .x.  H ) `  x )  =  ( ( G `  x
)  .x.  ( H `  x ) ) )
2215, 18, 19, 20, 21syl22anc 1265 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( G  oF  .x.  H ) `  x )  =  ( ( G `  x
)  .x.  ( H `  x ) ) )
2312, 22sylan2 476 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  ( G supp  Y ) ) )  ->  ( ( G  oF  .x.  H
) `  x )  =  ( ( G `
 x )  .x.  ( H `  x ) ) )
24 ssid 3489 . . . . . . . 8  |-  ( G supp 
Y )  C_  ( G supp  Y )
2524a1i 11 . . . . . . 7  |-  ( ph  ->  ( G supp  Y ) 
C_  ( G supp  Y
) )
26 lcomfsupp.y . . . . . . . . 9  |-  Y  =  ( 0g `  F
)
27 fvex 5891 . . . . . . . . 9  |-  ( 0g
`  F )  e. 
_V
2826, 27eqeltri 2513 . . . . . . . 8  |-  Y  e. 
_V
2928a1i 11 . . . . . . 7  |-  ( ph  ->  Y  e.  _V )
308, 25, 10, 29suppssr 6957 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  ( G supp  Y ) ) )  ->  ( G `  x )  =  Y )
3130oveq1d 6320 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  ( G supp  Y ) ) )  ->  ( ( G `
 x )  .x.  ( H `  x ) )  =  ( Y 
.x.  ( H `  x ) ) )
327adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  W  e.  LMod )
339ffvelrnda 6037 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  B )
34 lcomfsupp.z . . . . . . . 8  |-  .0.  =  ( 0g `  W )
356, 3, 5, 26, 34lmod0vs 18059 . . . . . . 7  |-  ( ( W  e.  LMod  /\  ( H `  x )  e.  B )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3632, 33, 35syl2anc 665 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3712, 36sylan2 476 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  ( G supp  Y ) ) )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3823, 31, 373eqtrd 2474 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  ( G supp  Y ) ) )  ->  ( ( G  oF  .x.  H
) `  x )  =  .0.  )
3911, 38suppss 6956 . . 3  |-  ( ph  ->  ( ( G  oF  .x.  H ) supp  .0.  )  C_  ( G supp  Y
) )
40 ssfi 7798 . . 3  |-  ( ( ( G supp  Y )  e.  Fin  /\  (
( G  oF  .x.  H ) supp  .0.  )  C_  ( G supp  Y
) )  ->  (
( G  oF  .x.  H ) supp  .0.  )  e.  Fin )
412, 39, 40syl2anc 665 . 2  |-  ( ph  ->  ( ( G  oF  .x.  H ) supp  .0.  )  e.  Fin )
42 inidm 3677 . . . . 5  |-  ( I  i^i  I )  =  I
4314, 17, 10, 10, 42offn 6556 . . . 4  |-  ( ph  ->  ( G  oF  .x.  H )  Fn  I )
44 fnfun 5691 . . . 4  |-  ( ( G  oF  .x.  H )  Fn  I  ->  Fun  ( G  oF  .x.  H ) )
4543, 44syl 17 . . 3  |-  ( ph  ->  Fun  ( G  oF  .x.  H ) )
46 ovex 6333 . . . 4  |-  ( G  oF  .x.  H
)  e.  _V
4746a1i 11 . . 3  |-  ( ph  ->  ( G  oF  .x.  H )  e. 
_V )
48 fvex 5891 . . . . 5  |-  ( 0g
`  W )  e. 
_V
4934, 48eqeltri 2513 . . . 4  |-  .0.  e.  _V
5049a1i 11 . . 3  |-  ( ph  ->  .0.  e.  _V )
51 funisfsupp 7894 . . 3  |-  ( ( Fun  ( G  oF  .x.  H )  /\  ( G  oF  .x.  H )  e.  _V  /\  .0.  e.  _V )  ->  ( ( G  oF  .x.  H ) finSupp  .0.  <->  (
( G  oF  .x.  H ) supp  .0.  )  e.  Fin )
)
5245, 47, 50, 51syl3anc 1264 . 2  |-  ( ph  ->  ( ( G  oF  .x.  H ) finSupp  .0.  <->  (
( G  oF  .x.  H ) supp  .0.  )  e.  Fin )
)
5341, 52mpbird 235 1  |-  ( ph  ->  ( G  oF  .x.  H ) finSupp  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    \ cdif 3439    C_ wss 3442   class class class wbr 4426   Fun wfun 5595    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    oFcof 6543   supp csupp 6925   Fincfn 7577   finSupp cfsupp 7889   Basecbs 15084  Scalarcsca 15155   .scvsca 15156   0gc0g 15297   LModclmod 18026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-supp 6926  df-er 7371  df-en 7578  df-fin 7581  df-fsupp 7890  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-ring 17717  df-lmod 18028
This theorem is referenced by:  islindf4  19327
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