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Theorem lfl0f 33741
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 5867 . . . . 5  |-  ( 0g
`  D )  e. 
_V
31, 2eqeltri 2544 . . . 4  |-  .0.  e.  _V
43fconst 5762 . . 3  |-  ( V  X.  {  .0.  }
) : V --> {  .0.  }
5 lfl0f.d . . . . 5  |-  D  =  (Scalar `  W )
6 eqid 2460 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
75, 6, 1lmod0cl 17314 . . . 4  |-  ( W  e.  LMod  ->  .0.  e.  ( Base `  D )
)
87snssd 4165 . . 3  |-  ( W  e.  LMod  ->  {  .0.  } 
C_  ( Base `  D
) )
9 fss 5730 . . 3  |-  ( ( ( V  X.  {  .0.  } ) : V --> {  .0.  }  /\  {  .0.  }  C_  ( Base `  D ) )  -> 
( V  X.  {  .0.  } ) : V --> ( Base `  D )
)
104, 8, 9sylancr 663 . 2  |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  }
) : V --> ( Base `  D ) )
115lmodrng 17296 . . . . . . . . 9  |-  ( W  e.  LMod  ->  D  e. 
Ring )
1211ad2antrr 725 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  D  e.  Ring )
13 simplrl 759 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  r  e.  ( Base `  D
) )
14 eqid 2460 . . . . . . . . 9  |-  ( .r
`  D )  =  ( .r `  D
)
156, 14, 1rngrz 17016 . . . . . . . 8  |-  ( ( D  e.  Ring  /\  r  e.  ( Base `  D
) )  ->  (
r ( .r `  D )  .0.  )  =  .0.  )
1612, 13, 15syl2anc 661 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
r ( .r `  D )  .0.  )  =  .0.  )
1716oveq1d 6290 . . . . . 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 16984 . . . . . . . 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 15872 . . . . . . . 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 2460 . . . . . . . 8  |-  ( +g  `  D )  =  ( +g  `  D )
236, 22, 1grplid 15874 . . . . . . 7  |-  ( ( D  e.  Grp  /\  .0.  e.  ( Base `  D
) )  ->  (  .0.  ( +g  `  D
)  .0.  )  =  .0.  )
2419, 21, 23syl2anc 661 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (  .0.  ( +g  `  D
)  .0.  )  =  .0.  )
2517, 24eqtrd 2501 . . . . 5  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( r ( .r
`  D )  .0.  ) ( +g  `  D
)  .0.  )  =  .0.  )
26 simplrr 760 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  x  e.  V )
273fvconst2 6107 . . . . . . . 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 6291 . . . . . 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 6107 . . . . . . 7  |-  ( y  e.  V  ->  (
( V  X.  {  .0.  } ) `  y
)  =  .0.  )
3130adantl 466 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( V  X.  {  .0.  } ) `  y
)  =  .0.  )
3229, 31oveq12d 6293 . . . . 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 753 . . . . . . 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 2460 . . . . . . . . 9  |-  ( .s
`  W )  =  ( .s `  W
)
3634, 5, 35, 6lmodvscl 17305 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  D
)  /\  x  e.  V )  ->  (
r ( .s `  W ) x )  e.  V )
3733, 13, 26, 36syl3anc 1223 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
r ( .s `  W ) x )  e.  V )
38 simpr 461 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  y  e.  V )
39 eqid 2460 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
4034, 39lmodvacl 17302 . . . . . . 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 1223 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( r ( .s
`  W ) x ) ( +g  `  W
) y )  e.  V )
423fvconst2 6107 . . . . . 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 2512 . . . 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 2871 . . 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 2878 . 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 33732 . 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 915 1  |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  }
)  e.  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   _Vcvv 3106    C_ wss 3469   {csn 4020    X. cxp 4990   -->wf 5575   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   .rcmulr 14545  Scalarcsca 14547   .scvsca 14548   0gc0g 14684   Grpcgrp 15716   Ringcrg 16979   LModclmod 17288  LFnlclfn 33729
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-0g 14686  df-mnd 15721  df-grp 15851  df-mgp 16925  df-rng 16981  df-lmod 17290  df-lfl 33730
This theorem is referenced by:  lkr0f  33766  lkrscss  33770  ldualgrplem  33817  ldual0v  33822  ldual0vcl  33823  lclkrlem1  36178  lclkr  36205  lclkrs  36211
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