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Theorem scmfsupp 32701
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 5611 . . 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 7835 . . 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 32700 . . 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 1262 . 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 6124 . . . . 5  |-  ( V  e.  ~P ( Base `  M )  ->  (
v  e.  V  |->  ( ( A `  v
) ( .s `  M ) v ) )  e.  _V )
109adantl 466 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  (
v  e.  V  |->  ( ( A `  v
) ( .s `  M ) v ) )  e.  _V )
11103ad2ant1 1016 . . 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 5863 . . 3  |-  ( 0g
`  M )  e. 
_V
13 isfsupp 7832 . . 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 662 . 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 920 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 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   _Vcvv 3093   ~Pcpw 3994   class class class wbr 4434    |-> cmpt 4492   Fun wfun 5569   ` cfv 5575  (class class class)co 6278   supp csupp 6900    ^m cmap 7419   Fincfn 7515   finSupp cfsupp 7828   Basecbs 14506  Scalarcsca 14574   .scvsca 14575   0gc0g 14711   LModclmod 17383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-supp 6901  df-er 7310  df-map 7421  df-en 7516  df-fin 7519  df-fsupp 7829  df-0g 14713  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-grp 15928  df-ring 17071  df-lmod 17385
This theorem is referenced by:  gsumlsscl  32706  lincfsuppcl  32744  linccl  32745  lincdifsn  32755  lincsum  32760  lincscm  32761  lincresunit3lem2  32811  lincresunit3  32812
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