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Theorem lfl0f 29259
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 5539 . . . . 5  |-  ( 0g
`  D )  e. 
_V
31, 2eqeltri 2353 . . . 4  |-  .0.  e.  _V
43fconst 5427 . . 3  |-  ( V  X.  {  .0.  }
) : V --> {  .0.  }
5 lfl0f.d . . . . 5  |-  D  =  (Scalar `  W )
6 eqid 2283 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
75, 6, 1lmod0cl 15656 . . . 4  |-  ( W  e.  LMod  ->  .0.  e.  ( Base `  D )
)
87snssd 3760 . . 3  |-  ( W  e.  LMod  ->  {  .0.  } 
C_  ( Base `  D
) )
9 fss 5397 . . 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 15635 . . . . . . . . 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 2283 . . . . . . . . 9  |-  ( .r
`  D )  =  ( .r `  D
)
156, 14, 1rngrz 15378 . . . . . . . 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 5873 . . . . . 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 15346 . . . . . . . 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 14510 . . . . . . . 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 2283 . . . . . . . 8  |-  ( +g  `  D )  =  ( +g  `  D )
236, 22, 1grplid 14512 . . . . . . 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 2315 . . . . 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 5729 . . . . . . . 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 5874 . . . . . 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 5729 . . . . . . 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 5876 . . . . 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 2283 . . . . . . . . 9  |-  ( .s
`  W )  =  ( .s `  W
)
3634, 5, 35, 6lmodvscl 15644 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  D
)  /\  x  e.  V )  ->  (
r ( .s `  W ) x )  e.  V )
3733, 13, 26, 36syl3anc 1182 . . . . . . 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 2283 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
4034, 39lmodvacl 15641 . . . . . . 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 1182 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  ( r  e.  (
Base `  D )  /\  x  e.  V
) )  /\  y  e.  V )  ->  (
( r ( .s
`  W ) x ) ( +g  `  W
) y )  e.  V )
423fvconst2 5729 . . . . . 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 2326 . . . 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 2626 . . 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 2635 . 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 29250 . 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   {csn 3640    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   Grpcgrp 14362   Ringcrg 15337   LModclmod 15627  LFnlclfn 29247
This theorem is referenced by:  lkr0f  29284  lkrscss  29288  ldualgrplem  29335  ldual0v  29340  ldual0vcl  29341  lclkrlem1  31696  lclkr  31723  lclkrs  31729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-mgp 15326  df-rng 15340  df-lmod 15629  df-lfl 29248
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