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Theorem lfl0f 29330
Description: The zero function is a functional. (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
lfl0f.d  |-  D  =  (Scalar `  W )
lfl0f.o  |-  .0.  =  ( 0g `  D )
lfl0f.v  |-  V  =  ( Base `  W
)
lfl0f.f  |-  F  =  (LFnl `  W )
Assertion
Ref Expression
lfl0f  |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  }
)  e.  F )

Proof of Theorem lfl0f
Dummy variables  x  r  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lfl0f.o . . . . 5  |-  .0.  =  ( 0g `  D )
2 fvex 5646 . . . . 5  |-  ( 0g
`  D )  e. 
_V
31, 2eqeltri 2436 . . . 4  |-  .0.  e.  _V
43fconst 5533 . . 3  |-  ( V  X.  {  .0.  }
) : V --> {  .0.  }
5 lfl0f.d . . . . 5  |-  D  =  (Scalar `  W )
6 eqid 2366 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
75, 6, 1lmod0cl 15866 . . . 4  |-  ( W  e.  LMod  ->  .0.  e.  ( Base `  D )
)
87snssd 3858 . . 3  |-  ( W  e.  LMod  ->  {  .0.  } 
C_  ( Base `  D
) )
9 fss 5503 . . 3  |-  ( ( ( V  X.  {  .0.  } ) : V --> {  .0.  }  /\  {  .0.  }  C_  ( Base `  D ) )  -> 
( V  X.  {  .0.  } ) : V --> ( Base `  D )
)
104, 8, 9sylancr 644 . 2  |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  }
) : V --> ( Base `  D ) )
115lmodrng 15845 . . . . . . . . 9  |-  ( W  e.  LMod  ->  D  e. 
Ring )
1211ad2antrr 706 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  D  e.  Ring )
13 simplrl 736 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  r  e.  ( Base `  D
) )
14 eqid 2366 . . . . . . . . 9  |-  ( .r
`  D )  =  ( .r `  D
)
156, 14, 1rngrz 15588 . . . . . . . 8  |-  ( ( D  e.  Ring  /\  r  e.  ( Base `  D
) )  ->  (
r ( .r `  D )  .0.  )  =  .0.  )
1612, 13, 15syl2anc 642 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
r ( .r `  D )  .0.  )  =  .0.  )
1716oveq1d 5996 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( r ( .r
`  D )  .0.  ) ( +g  `  D
)  .0.  )  =  (  .0.  ( +g  `  D )  .0.  )
)
18 rnggrp 15556 . . . . . . . 8  |-  ( D  e.  Ring  ->  D  e. 
Grp )
1912, 18syl 15 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  D  e.  Grp )
206, 1grpidcl 14720 . . . . . . . 8  |-  ( D  e.  Grp  ->  .0.  e.  ( Base `  D
) )
2119, 20syl 15 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  .0.  e.  ( Base `  D
) )
22 eqid 2366 . . . . . . . 8  |-  ( +g  `  D )  =  ( +g  `  D )
236, 22, 1grplid 14722 . . . . . . 7  |-  ( ( D  e.  Grp  /\  .0.  e.  ( Base `  D
) )  ->  (  .0.  ( +g  `  D
)  .0.  )  =  .0.  )
2419, 21, 23syl2anc 642 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (  .0.  ( +g  `  D
)  .0.  )  =  .0.  )
2517, 24eqtrd 2398 . . . . 5  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( r ( .r
`  D )  .0.  ) ( +g  `  D
)  .0.  )  =  .0.  )
26 simplrr 737 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  x  e.  V )
273fvconst2 5847 . . . . . . . 8  |-  ( x  e.  V  ->  (
( V  X.  {  .0.  } ) `  x
)  =  .0.  )
2826, 27syl 15 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( V  X.  {  .0.  } ) `  x
)  =  .0.  )
2928oveq2d 5997 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
r ( .r `  D ) ( ( V  X.  {  .0.  } ) `  x ) )  =  ( r ( .r `  D
)  .0.  ) )
303fvconst2 5847 . . . . . . 7  |-  ( y  e.  V  ->  (
( V  X.  {  .0.  } ) `  y
)  =  .0.  )
3130adantl 452 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( V  X.  {  .0.  } ) `  y
)  =  .0.  )
3229, 31oveq12d 5999 . . . . 5  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( r ( .r
`  D ) ( ( V  X.  {  .0.  } ) `  x
) ) ( +g  `  D ) ( ( V  X.  {  .0.  } ) `  y ) )  =  ( ( r ( .r `  D )  .0.  )
( +g  `  D )  .0.  ) )
33 simpll 730 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  W  e.  LMod )
34 lfl0f.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
35 eqid 2366 . . . . . . . . 9  |-  ( .s
`  W )  =  ( .s `  W
)
3634, 5, 35, 6lmodvscl 15854 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  D
)  /\  x  e.  V )  ->  (
r ( .s `  W ) x )  e.  V )
3733, 13, 26, 36syl3anc 1183 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
r ( .s `  W ) x )  e.  V )
38 simpr 447 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  y  e.  V )
39 eqid 2366 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
4034, 39lmodvacl 15851 . . . . . . 7  |-  ( ( W  e.  LMod  /\  (
r ( .s `  W ) x )  e.  V  /\  y  e.  V )  ->  (
( r ( .s
`  W ) x ) ( +g  `  W
) y )  e.  V )
4133, 37, 38, 40syl3anc 1183 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( r ( .s
`  W ) x ) ( +g  `  W
) y )  e.  V )
423fvconst2 5847 . . . . . 6  |-  ( ( ( r ( .s
`  W ) x ) ( +g  `  W
) y )  e.  V  ->  ( ( V  X.  {  .0.  }
) `  ( (
r ( .s `  W ) x ) ( +g  `  W
) y ) )  =  .0.  )
4341, 42syl 15 . . . . 5  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( V  X.  {  .0.  } ) `  (
( r ( .s
`  W ) x ) ( +g  `  W
) y ) )  =  .0.  )
4425, 32, 433eqtr4rd 2409 . . . 4  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( V  X.  {  .0.  } ) `  (
( r ( .s
`  W ) x ) ( +g  `  W
) y ) )  =  ( ( r ( .r `  D
) ( ( V  X.  {  .0.  }
) `  x )
) ( +g  `  D
) ( ( V  X.  {  .0.  }
) `  y )
) )
4544ralrimiva 2711 . . 3  |-  ( ( W  e.  LMod  /\  (
r  e.  ( Base `  D )  /\  x  e.  V ) )  ->  A. y  e.  V  ( ( V  X.  {  .0.  } ) `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  =  ( ( r ( .r `  D ) ( ( V  X.  {  .0.  } ) `  x ) ) ( +g  `  D
) ( ( V  X.  {  .0.  }
) `  y )
) )
4645ralrimivva 2720 . 2  |-  ( W  e.  LMod  ->  A. r  e.  ( Base `  D
) A. x  e.  V  A. y  e.  V  ( ( V  X.  {  .0.  }
) `  ( (
r ( .s `  W ) x ) ( +g  `  W
) y ) )  =  ( ( r ( .r `  D
) ( ( V  X.  {  .0.  }
) `  x )
) ( +g  `  D
) ( ( V  X.  {  .0.  }
) `  y )
) )
47 lfl0f.f . . 3  |-  F  =  (LFnl `  W )
4834, 39, 5, 35, 6, 22, 14, 47islfl 29321 . 2  |-  ( W  e.  LMod  ->  ( ( V  X.  {  .0.  } )  e.  F  <->  ( ( V  X.  {  .0.  }
) : V --> ( Base `  D )  /\  A. r  e.  ( Base `  D ) A. x  e.  V  A. y  e.  V  ( ( V  X.  {  .0.  }
) `  ( (
r ( .s `  W ) x ) ( +g  `  W
) y ) )  =  ( ( r ( .r `  D
) ( ( V  X.  {  .0.  }
) `  x )
) ( +g  `  D
) ( ( V  X.  {  .0.  }
) `  y )
) ) ) )
4910, 46, 48mpbir2and 888 1  |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  }
)  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   _Vcvv 2873    C_ wss 3238   {csn 3729    X. cxp 4790   -->wf 5354   ` cfv 5358  (class class class)co 5981   Basecbs 13356   +g cplusg 13416   .rcmulr 13417  Scalarcsca 13419   .scvsca 13420   0gc0g 13610   Grpcgrp 14572   Ringcrg 15547   LModclmod 15837  LFnlclfn 29318
This theorem is referenced by:  lkr0f  29355  lkrscss  29359  ldualgrplem  29406  ldual0v  29411  ldual0vcl  29412  lclkrlem1  31767  lclkr  31794  lclkrs  31800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-plusg 13429  df-0g 13614  df-mnd 14577  df-grp 14699  df-mgp 15536  df-rng 15550  df-lmod 15839  df-lfl 29319
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