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Theorem lkr0f 32112
Description: The kernel of the zero functional is the set of all vectors. (Contributed by NM, 17-Apr-2014.)
Hypotheses
Ref Expression
lkr0f.d  |-  D  =  (Scalar `  W )
lkr0f.o  |-  .0.  =  ( 0g `  D )
lkr0f.v  |-  V  =  ( Base `  W
)
lkr0f.f  |-  F  =  (LFnl `  W )
lkr0f.k  |-  K  =  (LKer `  W )
Assertion
Ref Expression
lkr0f  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( K `  G
)  =  V  <->  G  =  ( V  X.  {  .0.  } ) ) )

Proof of Theorem lkr0f
StepHypRef Expression
1 lkr0f.d . . . . . . 7  |-  D  =  (Scalar `  W )
2 eqid 2402 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
3 lkr0f.v . . . . . . 7  |-  V  =  ( Base `  W
)
4 lkr0f.f . . . . . . 7  |-  F  =  (LFnl `  W )
51, 2, 3, 4lflf 32081 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> ( Base `  D
) )
6 ffn 5714 . . . . . 6  |-  ( G : V --> ( Base `  D )  ->  G  Fn  V )
75, 6syl 17 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G  Fn  V )
87adantr 463 . . . 4  |-  ( ( ( W  e.  LMod  /\  G  e.  F )  /\  ( K `  G )  =  V )  ->  G  Fn  V )
9 lkr0f.o . . . . . . 7  |-  .0.  =  ( 0g `  D )
10 lkr0f.k . . . . . . 7  |-  K  =  (LKer `  W )
111, 9, 4, 10lkrval 32106 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
1211eqeq1d 2404 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( K `  G
)  =  V  <->  ( `' G " {  .0.  }
)  =  V ) )
1312biimpa 482 . . . 4  |-  ( ( ( W  e.  LMod  /\  G  e.  F )  /\  ( K `  G )  =  V )  ->  ( `' G " {  .0.  }
)  =  V )
14 fvex 5859 . . . . . . 7  |-  ( 0g
`  D )  e. 
_V
159, 14eqeltri 2486 . . . . . 6  |-  .0.  e.  _V
1615fconst2 6108 . . . . 5  |-  ( G : V --> {  .0.  }  <-> 
G  =  ( V  X.  {  .0.  }
) )
17 fconst4 6117 . . . . 5  |-  ( G : V --> {  .0.  }  <-> 
( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) )
1816, 17bitr3i 251 . . . 4  |-  ( G  =  ( V  X.  {  .0.  } )  <->  ( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) )
198, 13, 18sylanbrc 662 . . 3  |-  ( ( ( W  e.  LMod  /\  G  e.  F )  /\  ( K `  G )  =  V )  ->  G  =  ( V  X.  {  .0.  } ) )
2019ex 432 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( K `  G
)  =  V  ->  G  =  ( V  X.  {  .0.  } ) ) )
2118biimpi 194 . . . . . 6  |-  ( G  =  ( V  X.  {  .0.  } )  -> 
( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) )
2221adantl 464 . . . . 5  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) )
23 simpr 459 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  G  =  ( V  X.  {  .0.  } ) )
24 eqid 2402 . . . . . . . . . . 11  |-  (LFnl `  W )  =  (LFnl `  W )
251, 9, 3, 24lfl0f 32087 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  }
)  e.  (LFnl `  W ) )
2625adantr 463 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( V  X.  {  .0.  } )  e.  (LFnl `  W )
)
2723, 26eqeltrd 2490 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  G  e.  (LFnl `  W ) )
281, 9, 24, 10lkrval 32106 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  (LFnl `  W )
)  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
2927, 28syldan 468 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
3029eqeq1d 2404 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( ( K `
 G )  =  V  <->  ( `' G " {  .0.  } )  =  V ) )
31 ffn 5714 . . . . . . . . 9  |-  ( G : V --> {  .0.  }  ->  G  Fn  V
)
3216, 31sylbir 213 . . . . . . . 8  |-  ( G  =  ( V  X.  {  .0.  } )  ->  G  Fn  V )
3332adantl 464 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  G  Fn  V
)
3433biantrurd 506 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( ( `' G " {  .0.  } )  =  V  <->  ( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) ) )
3530, 34bitrd 253 . . . . 5  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( ( K `
 G )  =  V  <->  ( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) ) )
3622, 35mpbird 232 . . . 4  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( K `  G )  =  V )
3736ex 432 . . 3  |-  ( W  e.  LMod  ->  ( G  =  ( V  X.  {  .0.  } )  -> 
( K `  G
)  =  V ) )
3837adantr 463 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( G  =  ( V  X.  {  .0.  } )  ->  ( K `  G )  =  V ) )
3920, 38impbid 190 1  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( K `  G
)  =  V  <->  G  =  ( V  X.  {  .0.  } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3059   {csn 3972    X. cxp 4821   `'ccnv 4822   "cima 4826    Fn wfn 5564   -->wf 5565   ` cfv 5569   Basecbs 14841  Scalarcsca 14912   0gc0g 15054   LModclmod 17832  LFnlclfn 32075  LKerclk 32103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-plusg 14922  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381  df-mgp 17462  df-ring 17520  df-lmod 17834  df-lfl 32076  df-lkr 32104
This theorem is referenced by:  lkrscss  32116  eqlkr  32117  lkrshp  32123  lkrshp3  32124  lkrshpor  32125  lfl1dim  32139  lfl1dim2N  32140  lkr0f2  32179  lclkrlem1  34526  lclkrlem2j  34536  lclkr  34553  lclkrs  34559  mapd0  34685
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