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Theorem lcomfsupp 16985
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 7627 . . 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 16983 . . . 4  |-  ( ph  ->  ( G  oF  .x.  H ) : I --> B )
12 eldifi 3478 . . . . . 6  |-  ( x  e.  ( I  \ 
( G supp  Y )
)  ->  x  e.  I )
13 ffn 5559 . . . . . . . . 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 5559 . . . . . . . . 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 6333 . . . . . . 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 1219 . . . . . 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 3375 . . . . . . . 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 5701 . . . . . . . . 9  |-  ( 0g
`  F )  e. 
_V
2826, 27eqeltri 2513 . . . . . . . 8  |-  Y  e. 
_V
2928a1i 11 . . . . . . 7  |-  ( ph  ->  Y  e.  _V )
308, 25, 10, 29suppssr 6720 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  ( G supp  Y ) ) )  ->  ( G `  x )  =  Y )
3130oveq1d 6106 . . . . 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 5843 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  B )
34 lcomfsupp.z . . . . . . . 8  |-  .0.  =  ( 0g `  W )
356, 3, 5, 26, 34lmod0vs 16981 . . . . . . 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 2479 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  ( G supp  Y ) ) )  ->  ( ( G  oF  .x.  H
) `  x )  =  .0.  )
3911, 38suppss 6719 . . 3  |-  ( ph  ->  ( ( G  oF  .x.  H ) supp  .0.  )  C_  ( G supp  Y
) )
40 ssfi 7533 . . 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 3559 . . . . 5  |-  ( I  i^i  I )  =  I
4314, 17, 10, 10, 42offn 6331 . . . 4  |-  ( ph  ->  ( G  oF  .x.  H )  Fn  I )
44 fnfun 5508 . . . 4  |-  ( ( G  oF  .x.  H )  Fn  I  ->  Fun  ( G  oF  .x.  H ) )
4543, 44syl 16 . . 3  |-  ( ph  ->  Fun  ( G  oF  .x.  H ) )
46 ovex 6116 . . . 4  |-  ( G  oF  .x.  H
)  e.  _V
4746a1i 11 . . 3  |-  ( ph  ->  ( G  oF  .x.  H )  e. 
_V )
48 fvex 5701 . . . . 5  |-  ( 0g
`  W )  e. 
_V
4934, 48eqeltri 2513 . . . 4  |-  .0.  e.  _V
5049a1i 11 . . 3  |-  ( ph  ->  .0.  e.  _V )
51 funisfsupp 7625 . . 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 1218 . 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 1369    e. wcel 1756   _Vcvv 2972    \ cdif 3325    C_ wss 3328   class class class wbr 4292   Fun wfun 5412    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091    oFcof 6318   supp csupp 6690   Fincfn 7310   finSupp cfsupp 7620   Basecbs 14174  Scalarcsca 14241   .scvsca 14242   0gc0g 14378   LModclmod 16948
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-supp 6691  df-er 7101  df-en 7311  df-fin 7314  df-fsupp 7621  df-0g 14380  df-mnd 15415  df-grp 15545  df-rng 16647  df-lmod 16950
This theorem is referenced by:  islindf4  18267
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