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Theorem lfl0f 34496
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 5862 . . . . 5  |-  ( 0g
`  D )  e. 
_V
31, 2eqeltri 2525 . . . 4  |-  .0.  e.  _V
43fconst 5757 . . 3  |-  ( V  X.  {  .0.  }
) : V --> {  .0.  }
5 lfl0f.d . . . . 5  |-  D  =  (Scalar `  W )
6 eqid 2441 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
75, 6, 1lmod0cl 17406 . . . 4  |-  ( W  e.  LMod  ->  .0.  e.  ( Base `  D )
)
87snssd 4156 . . 3  |-  ( W  e.  LMod  ->  {  .0.  } 
C_  ( Base `  D
) )
9 fss 5725 . . 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 ) )
115lmodring 17388 . . . . . . . . 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 2441 . . . . . . . . 9  |-  ( .r
`  D )  =  ( .r `  D
)
156, 14, 1ringrz 17104 . . . . . . . 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 6292 . . . . . 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 ringgrp 17071 . . . . . . . 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 15947 . . . . . . . 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 15949 . . . . . . 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 2482 . . . . 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 6293 . . . . . 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 6295 . . . . 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 2441 . . . . . . . . 9  |-  ( .s
`  W )  =  ( .s `  W
)
3634, 5, 35, 6lmodvscl 17397 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  D
)  /\  x  e.  V )  ->  (
r ( .s `  W ) x )  e.  V )
3733, 13, 26, 36syl3anc 1227 . . . . . . 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 2441 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
4034, 39lmodvacl 17394 . . . . . . 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 1227 . . . . . 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 2493 . . . 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 2855 . . 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 2862 . 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 34487 . 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 920 1  |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  }
)  e.  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   A.wral 2791   _Vcvv 3093    C_ wss 3458   {csn 4010    X. cxp 4983   -->wf 5570   ` cfv 5574  (class class class)co 6277   Basecbs 14504   +g cplusg 14569   .rcmulr 14570  Scalarcsca 14572   .scvsca 14573   0gc0g 14709   Grpcgrp 15922   Ringcrg 17066   LModclmod 17380  LFnlclfn 34484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-plusg 14582  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-mgp 17010  df-ring 17068  df-lmod 17382  df-lfl 34485
This theorem is referenced by:  lkr0f  34521  lkrscss  34525  ldualgrplem  34572  ldual0v  34577  ldual0vcl  34578  lclkrlem1  36935  lclkr  36962  lclkrs  36968
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