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Theorem scmfsupp 32070
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 5624 . . 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 7836 . . 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 32069 . . 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 1263 . 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 6130 . . . . 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 1017 . . 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 5876 . . 3  |-  ( 0g
`  M )  e. 
_V
13 isfsupp 7833 . . 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   ~Pcpw 4010   class class class wbr 4447    |-> cmpt 4505   Fun wfun 5582   ` cfv 5588  (class class class)co 6284   supp csupp 6901    ^m cmap 7420   Fincfn 7516   finSupp cfsupp 7829   Basecbs 14490  Scalarcsca 14558   .scvsca 14559   0gc0g 14695   LModclmod 17312
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 6576
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-er 7311  df-map 7422  df-en 7517  df-fin 7520  df-fsupp 7830  df-0g 14697  df-mnd 15732  df-grp 15867  df-rng 17002  df-lmod 17314
This theorem is referenced by:  gsumlsscl  32075  lincfsuppcl  32113  linccl  32114  lincdifsn  32124  lincsum  32129  lincscm  32130  lincresunit3lem2  32180  lincresunit3  32181
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