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Theorem lcomfsupp 17345
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 7835 . . 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 17343 . . . 4  |-  ( ph  ->  ( G  oF  .x.  H ) : I --> B )
12 eldifi 3626 . . . . . 6  |-  ( x  e.  ( I  \ 
( G supp  Y )
)  ->  x  e.  I )
13 ffn 5730 . . . . . . . . 9  |-  ( G : I --> K  ->  G  Fn  I )
148, 13syl 16 . . . . . . . 8  |-  ( ph  ->  G  Fn  I )
1514adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  G  Fn  I )
16 ffn 5730 . . . . . . . . 9  |-  ( H : I --> B  ->  H  Fn  I )
179, 16syl 16 . . . . . . . 8  |-  ( ph  ->  H  Fn  I )
1817adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  H  Fn  I )
1910adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  V )
20 simpr 461 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
21 fnfvof 6536 . . . . . . 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 1229 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( G  oF  .x.  H ) `  x )  =  ( ( G `  x
)  .x.  ( H `  x ) ) )
2312, 22sylan2 474 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  ( G supp  Y ) ) )  ->  ( ( G  oF  .x.  H
) `  x )  =  ( ( G `
 x )  .x.  ( H `  x ) ) )
24 ssid 3523 . . . . . . . 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 5875 . . . . . . . . 9  |-  ( 0g
`  F )  e. 
_V
2826, 27eqeltri 2551 . . . . . . . 8  |-  Y  e. 
_V
2928a1i 11 . . . . . . 7  |-  ( ph  ->  Y  e.  _V )
308, 25, 10, 29suppssr 6931 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  ( G supp  Y ) ) )  ->  ( G `  x )  =  Y )
3130oveq1d 6298 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  ( G supp  Y ) ) )  ->  ( ( G `
 x )  .x.  ( H `  x ) )  =  ( Y 
.x.  ( H `  x ) ) )
327adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  W  e.  LMod )
339ffvelrnda 6020 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  B )
34 lcomfsupp.z . . . . . . . 8  |-  .0.  =  ( 0g `  W )
356, 3, 5, 26, 34lmod0vs 17340 . . . . . . 7  |-  ( ( W  e.  LMod  /\  ( H `  x )  e.  B )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3632, 33, 35syl2anc 661 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3712, 36sylan2 474 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  ( G supp  Y ) ) )  ->  ( Y  .x.  ( H `  x ) )  =  .0.  )
3823, 31, 373eqtrd 2512 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  ( G supp  Y ) ) )  ->  ( ( G  oF  .x.  H
) `  x )  =  .0.  )
3911, 38suppss 6930 . . 3  |-  ( ph  ->  ( ( G  oF  .x.  H ) supp  .0.  )  C_  ( G supp  Y
) )
40 ssfi 7740 . . 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 661 . 2  |-  ( ph  ->  ( ( G  oF  .x.  H ) supp  .0.  )  e.  Fin )
42 inidm 3707 . . . . 5  |-  ( I  i^i  I )  =  I
4314, 17, 10, 10, 42offn 6534 . . . 4  |-  ( ph  ->  ( G  oF  .x.  H )  Fn  I )
44 fnfun 5677 . . . 4  |-  ( ( G  oF  .x.  H )  Fn  I  ->  Fun  ( G  oF  .x.  H ) )
4543, 44syl 16 . . 3  |-  ( ph  ->  Fun  ( G  oF  .x.  H ) )
46 ovex 6308 . . . 4  |-  ( G  oF  .x.  H
)  e.  _V
4746a1i 11 . . 3  |-  ( ph  ->  ( G  oF  .x.  H )  e. 
_V )
48 fvex 5875 . . . . 5  |-  ( 0g
`  W )  e. 
_V
4934, 48eqeltri 2551 . . . 4  |-  .0.  e.  _V
5049a1i 11 . . 3  |-  ( ph  ->  .0.  e.  _V )
51 funisfsupp 7833 . . 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 1228 . 2  |-  ( ph  ->  ( ( G  oF  .x.  H ) finSupp  .0.  <->  (
( G  oF  .x.  H ) supp  .0.  )  e.  Fin )
)
5341, 52mpbird 232 1  |-  ( ph  ->  ( G  oF  .x.  H ) finSupp  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    \ cdif 3473    C_ wss 3476   class class class wbr 4447   Fun wfun 5581    Fn wfn 5582   -->wf 5583   ` cfv 5587  (class class class)co 6283    oFcof 6521   supp csupp 6901   Fincfn 7516   finSupp cfsupp 7828   Basecbs 14489  Scalarcsca 14557   .scvsca 14558   0gc0g 14694   LModclmod 17307
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-supp 6902  df-er 7311  df-en 7517  df-fin 7520  df-fsupp 7829  df-0g 14696  df-mnd 15731  df-grp 15864  df-rng 16997  df-lmod 17309
This theorem is referenced by:  islindf4  18656
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