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Theorem lindslinindimp2lem3 39013
Description: Lemma 3 for lindslinindsimp2 39016. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Hypotheses
Ref Expression
lindslinind.r  |-  R  =  (Scalar `  M )
lindslinind.b  |-  B  =  ( Base `  R
)
lindslinind.0  |-  .0.  =  ( 0g `  R )
lindslinind.z  |-  Z  =  ( 0g `  M
)
lindslinind.y  |-  Y  =  ( ( invg `  R ) `  (
f `  x )
)
lindslinind.g  |-  G  =  ( f  |`  ( S  \  { x }
) )
Assertion
Ref Expression
lindslinindimp2lem3  |-  ( ( ( S  e.  V  /\  M  e.  LMod )  /\  ( S  C_  ( Base `  M )  /\  x  e.  S
)  /\  ( f  e.  ( B  ^m  S
)  /\  f finSupp  .0.  )
)  ->  G finSupp  .0.  )
Distinct variable groups:    B, f    f, M    R, f, x    S, f, x    f, Z    .0. , f, x
Allowed substitution hints:    B( x)    G( x, f)    M( x)    V( x, f)    Y( x, f)    Z( x)

Proof of Theorem lindslinindimp2lem3
StepHypRef Expression
1 lindslinind.g . 2  |-  G  =  ( f  |`  ( S  \  { x }
) )
2 simp3r 1034 . . 3  |-  ( ( ( S  e.  V  /\  M  e.  LMod )  /\  ( S  C_  ( Base `  M )  /\  x  e.  S
)  /\  ( f  e.  ( B  ^m  S
)  /\  f finSupp  .0.  )
)  ->  f finSupp  .0.  )
3 lindslinind.0 . . . . 5  |-  .0.  =  ( 0g `  R )
4 fvex 5891 . . . . 5  |-  ( 0g
`  R )  e. 
_V
53, 4eqeltri 2513 . . . 4  |-  .0.  e.  _V
65a1i 11 . . 3  |-  ( ( ( S  e.  V  /\  M  e.  LMod )  /\  ( S  C_  ( Base `  M )  /\  x  e.  S
)  /\  ( f  e.  ( B  ^m  S
)  /\  f finSupp  .0.  )
)  ->  .0.  e.  _V )
72, 6fsuppres 7914 . 2  |-  ( ( ( S  e.  V  /\  M  e.  LMod )  /\  ( S  C_  ( Base `  M )  /\  x  e.  S
)  /\  ( f  e.  ( B  ^m  S
)  /\  f finSupp  .0.  )
)  ->  ( f  |`  ( S  \  {
x } ) ) finSupp  .0.  )
81, 7syl5eqbr 4459 1  |-  ( ( ( S  e.  V  /\  M  e.  LMod )  /\  ( S  C_  ( Base `  M )  /\  x  e.  S
)  /\  ( f  e.  ( B  ^m  S
)  /\  f finSupp  .0.  )
)  ->  G finSupp  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   _Vcvv 3087    \ cdif 3439    C_ wss 3442   {csn 4002   class class class wbr 4426    |` cres 4856   ` cfv 5601  (class class class)co 6305    ^m cmap 7480   finSupp cfsupp 7889   Basecbs 15084  Scalarcsca 15155   0gc0g 15297   invgcminusg 16621   LModclmod 18026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
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 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-supp 6926  df-er 7371  df-en 7578  df-fin 7581  df-fsupp 7890
This theorem is referenced by:  lindslinindimp2lem4  39014
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