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Theorem lfl0f 32436
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 5698 . . . . 5  |-  ( 0g
`  D )  e. 
_V
31, 2eqeltri 2511 . . . 4  |-  .0.  e.  _V
43fconst 5593 . . 3  |-  ( V  X.  {  .0.  }
) : V --> {  .0.  }
5 lfl0f.d . . . . 5  |-  D  =  (Scalar `  W )
6 eqid 2441 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
75, 6, 1lmod0cl 16954 . . . 4  |-  ( W  e.  LMod  ->  .0.  e.  ( Base `  D )
)
87snssd 4015 . . 3  |-  ( W  e.  LMod  ->  {  .0.  } 
C_  ( Base `  D
) )
9 fss 5564 . . 3  |-  ( ( ( V  X.  {  .0.  } ) : V --> {  .0.  }  /\  {  .0.  }  C_  ( Base `  D ) )  -> 
( V  X.  {  .0.  } ) : V --> ( Base `  D )
)
104, 8, 9sylancr 658 . 2  |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  }
) : V --> ( Base `  D ) )
115lmodrng 16936 . . . . . . . . 9  |-  ( W  e.  LMod  ->  D  e. 
Ring )
1211ad2antrr 720 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  D  e.  Ring )
13 simplrl 754 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  r  e.  ( Base `  D
) )
14 eqid 2441 . . . . . . . . 9  |-  ( .r
`  D )  =  ( .r `  D
)
156, 14, 1rngrz 16672 . . . . . . . 8  |-  ( ( D  e.  Ring  /\  r  e.  ( Base `  D
) )  ->  (
r ( .r `  D )  .0.  )  =  .0.  )
1612, 13, 15syl2anc 656 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
r ( .r `  D )  .0.  )  =  .0.  )
1716oveq1d 6105 . . . . . 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 16640 . . . . . . . 8  |-  ( D  e.  Ring  ->  D  e. 
Grp )
1912, 18syl 16 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  D  e.  Grp )
206, 1grpidcl 15559 . . . . . . . 8  |-  ( D  e.  Grp  ->  .0.  e.  ( Base `  D
) )
2119, 20syl 16 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  .0.  e.  ( Base `  D
) )
22 eqid 2441 . . . . . . . 8  |-  ( +g  `  D )  =  ( +g  `  D )
236, 22, 1grplid 15561 . . . . . . 7  |-  ( ( D  e.  Grp  /\  .0.  e.  ( Base `  D
) )  ->  (  .0.  ( +g  `  D
)  .0.  )  =  .0.  )
2419, 21, 23syl2anc 656 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (  .0.  ( +g  `  D
)  .0.  )  =  .0.  )
2517, 24eqtrd 2473 . . . . 5  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( r ( .r
`  D )  .0.  ) ( +g  `  D
)  .0.  )  =  .0.  )
26 simplrr 755 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  x  e.  V )
273fvconst2 5930 . . . . . . . 8  |-  ( x  e.  V  ->  (
( V  X.  {  .0.  } ) `  x
)  =  .0.  )
2826, 27syl 16 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( V  X.  {  .0.  } ) `  x
)  =  .0.  )
2928oveq2d 6106 . . . . . 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 5930 . . . . . . 7  |-  ( y  e.  V  ->  (
( V  X.  {  .0.  } ) `  y
)  =  .0.  )
3130adantl 463 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( V  X.  {  .0.  } ) `  y
)  =  .0.  )
3229, 31oveq12d 6108 . . . . 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 748 . . . . . . 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 2441 . . . . . . . . 9  |-  ( .s
`  W )  =  ( .s `  W
)
3634, 5, 35, 6lmodvscl 16945 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  D
)  /\  x  e.  V )  ->  (
r ( .s `  W ) x )  e.  V )
3733, 13, 26, 36syl3anc 1213 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
r ( .s `  W ) x )  e.  V )
38 simpr 458 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  y  e.  V )
39 eqid 2441 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
4034, 39lmodvacl 16942 . . . . . . 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 1213 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( r ( .s
`  W ) x ) ( +g  `  W
) y )  e.  V )
423fvconst2 5930 . . . . . 6  |-  ( ( ( r ( .s
`  W ) x ) ( +g  `  W
) y )  e.  V  ->  ( ( V  X.  {  .0.  }
) `  ( (
r ( .s `  W ) x ) ( +g  `  W
) y ) )  =  .0.  )
4341, 42syl 16 . . . . 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 2484 . . . 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 2797 . . 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 2806 . 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 32427 . 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 908 1  |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  }
)  e.  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    C_ wss 3325   {csn 3874    X. cxp 4834   -->wf 5411   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   .rcmulr 14235  Scalarcsca 14237   .scvsca 14238   0gc0g 14374   Grpcgrp 15406   Ringcrg 16635   LModclmod 16928  LFnlclfn 32424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-0g 14376  df-mnd 15411  df-grp 15538  df-mgp 16582  df-rng 16637  df-lmod 16930  df-lfl 32425
This theorem is referenced by:  lkr0f  32461  lkrscss  32465  ldualgrplem  32512  ldual0v  32517  ldual0vcl  32518  lclkrlem1  34873  lclkr  34900  lclkrs  34906
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