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Theorem scmfsupp 39766
Description: A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
Hypotheses
Ref Expression
scmsuppfi.s  |-  S  =  (Scalar `  M )
scmsuppfi.r  |-  R  =  ( Base `  S
)
Assertion
Ref Expression
scmfsupp  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  A  e.  ( R  ^m  V )  /\  A finSupp  ( 0g `  S ) )  -> 
( v  e.  V  |->  ( ( A `  v ) ( .s
`  M ) v ) ) finSupp  ( 0g
`  M ) )
Distinct variable groups:    v, A    v, M    v, R    v, V
Allowed substitution hint:    S( v)

Proof of Theorem scmfsupp
StepHypRef Expression
1 funmpt 5580 . . 3  |-  Fun  (
v  e.  V  |->  ( ( A `  v
) ( .s `  M ) v ) )
21a1i 11 . 2  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  A  e.  ( R  ^m  V )  /\  A finSupp  ( 0g `  S ) )  ->  Fun  ( v  e.  V  |->  ( ( A `  v ) ( .s
`  M ) v ) ) )
3 id 22 . . . 4  |-  ( A finSupp 
( 0g `  S
)  ->  A finSupp  ( 0g
`  S ) )
43fsuppimpd 7843 . . 3  |-  ( A finSupp 
( 0g `  S
)  ->  ( A supp  ( 0g `  S ) )  e.  Fin )
5 scmsuppfi.s . . . 4  |-  S  =  (Scalar `  M )
6 scmsuppfi.r . . . 4  |-  R  =  ( Base `  S
)
75, 6scmsuppfi 39765 . . 3  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  A  e.  ( R  ^m  V )  /\  ( A supp  ( 0g `  S ) )  e.  Fin )  -> 
( ( v  e.  V  |->  ( ( A `
 v ) ( .s `  M ) v ) ) supp  ( 0g `  M ) )  e.  Fin )
84, 7syl3an3 1299 . 2  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  A  e.  ( R  ^m  V )  /\  A finSupp  ( 0g `  S ) )  -> 
( ( v  e.  V  |->  ( ( A `
 v ) ( .s `  M ) v ) ) supp  ( 0g `  M ) )  e.  Fin )
9 mptexg 6094 . . . . 5  |-  ( V  e.  ~P ( Base `  M )  ->  (
v  e.  V  |->  ( ( A `  v
) ( .s `  M ) v ) )  e.  _V )
109adantl 467 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  (
v  e.  V  |->  ( ( A `  v
) ( .s `  M ) v ) )  e.  _V )
11103ad2ant1 1026 . . 3  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  A  e.  ( R  ^m  V )  /\  A finSupp  ( 0g `  S ) )  -> 
( v  e.  V  |->  ( ( A `  v ) ( .s
`  M ) v ) )  e.  _V )
12 fvex 5835 . . 3  |-  ( 0g
`  M )  e. 
_V
13 isfsupp 7840 . . 3  |-  ( ( ( v  e.  V  |->  ( ( A `  v ) ( .s
`  M ) v ) )  e.  _V  /\  ( 0g `  M
)  e.  _V )  ->  ( ( v  e.  V  |->  ( ( A `
 v ) ( .s `  M ) v ) ) finSupp  ( 0g `  M )  <->  ( Fun  ( v  e.  V  |->  ( ( A `  v ) ( .s
`  M ) v ) )  /\  (
( v  e.  V  |->  ( ( A `  v ) ( .s
`  M ) v ) ) supp  ( 0g
`  M ) )  e.  Fin ) ) )
1411, 12, 13sylancl 666 . 2  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  A  e.  ( R  ^m  V )  /\  A finSupp  ( 0g `  S ) )  -> 
( ( v  e.  V  |->  ( ( A `
 v ) ( .s `  M ) v ) ) finSupp  ( 0g `  M )  <->  ( Fun  ( v  e.  V  |->  ( ( A `  v ) ( .s
`  M ) v ) )  /\  (
( v  e.  V  |->  ( ( A `  v ) ( .s
`  M ) v ) ) supp  ( 0g
`  M ) )  e.  Fin ) ) )
152, 8, 14mpbir2and 930 1  |-  ( ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) )  /\  A  e.  ( R  ^m  V )  /\  A finSupp  ( 0g `  S ) )  -> 
( v  e.  V  |->  ( ( A `  v ) ( .s
`  M ) v ) ) finSupp  ( 0g
`  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   _Vcvv 3022   ~Pcpw 3924   class class class wbr 4366    |-> cmpt 4425   Fun wfun 5538   ` cfv 5544  (class class class)co 6249   supp csupp 6869    ^m cmap 7427   Fincfn 7524   finSupp cfsupp 7836   Basecbs 15064  Scalarcsca 15136   .scvsca 15137   0gc0g 15281   LModclmod 18034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
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 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-supp 6870  df-er 7318  df-map 7429  df-en 7525  df-fin 7528  df-fsupp 7837  df-0g 15283  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-grp 16616  df-ring 17725  df-lmod 18036
This theorem is referenced by:  gsumlsscl  39771  lincfsuppcl  39809  linccl  39810  lincdifsn  39820  lincsum  39825  lincscm  39826  lincresunit3lem2  39876  lincresunit3  39877
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