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Theorem lkr0f 33891
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 2467 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
3 lkr0f.v . . . . . . 7  |-  V  =  ( Base `  W
)
4 lkr0f.f . . . . . . 7  |-  F  =  (LFnl `  W )
51, 2, 3, 4lflf 33860 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> ( Base `  D
) )
6 ffn 5729 . . . . . 6  |-  ( G : V --> ( Base `  D )  ->  G  Fn  V )
75, 6syl 16 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G  Fn  V )
87adantr 465 . . . 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 33885 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
1211eqeq1d 2469 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( K `  G
)  =  V  <->  ( `' G " {  .0.  }
)  =  V ) )
1312biimpa 484 . . . 4  |-  ( ( ( W  e.  LMod  /\  G  e.  F )  /\  ( K `  G )  =  V )  ->  ( `' G " {  .0.  }
)  =  V )
14 fvex 5874 . . . . . . 7  |-  ( 0g
`  D )  e. 
_V
159, 14eqeltri 2551 . . . . . 6  |-  .0.  e.  _V
1615fconst2 6115 . . . . 5  |-  ( G : V --> {  .0.  }  <-> 
G  =  ( V  X.  {  .0.  }
) )
17 fconst4 6123 . . . . 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 664 . . 3  |-  ( ( ( W  e.  LMod  /\  G  e.  F )  /\  ( K `  G )  =  V )  ->  G  =  ( V  X.  {  .0.  } ) )
2019ex 434 . 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 466 . . . . 5  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) )
23 simpr 461 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  G  =  ( V  X.  {  .0.  } ) )
24 eqid 2467 . . . . . . . . . . 11  |-  (LFnl `  W )  =  (LFnl `  W )
251, 9, 3, 24lfl0f 33866 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  }
)  e.  (LFnl `  W ) )
2625adantr 465 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( V  X.  {  .0.  } )  e.  (LFnl `  W )
)
2723, 26eqeltrd 2555 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  G  e.  (LFnl `  W ) )
281, 9, 24, 10lkrval 33885 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  (LFnl `  W )
)  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
2927, 28syldan 470 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
3029eqeq1d 2469 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( ( K `
 G )  =  V  <->  ( `' G " {  .0.  } )  =  V ) )
31 ffn 5729 . . . . . . . . 9  |-  ( G : V --> {  .0.  }  ->  G  Fn  V
)
3216, 31sylbir 213 . . . . . . . 8  |-  ( G  =  ( V  X.  {  .0.  } )  ->  G  Fn  V )
3332adantl 466 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  G  Fn  V
)
3433biantrurd 508 . . . . . 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 434 . . 3  |-  ( W  e.  LMod  ->  ( G  =  ( V  X.  {  .0.  } )  -> 
( K `  G
)  =  V ) )
3837adantr 465 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( G  =  ( V  X.  {  .0.  } )  ->  ( K `  G )  =  V ) )
3920, 38impbid 191 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 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027    X. cxp 4997   `'ccnv 4998   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586   Basecbs 14483  Scalarcsca 14551   0gc0g 14688   LModclmod 17292  LFnlclfn 33854  LKerclk 33882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-plusg 14561  df-0g 14690  df-mnd 15725  df-grp 15855  df-mgp 16929  df-rng 16985  df-lmod 17294  df-lfl 33855  df-lkr 33883
This theorem is referenced by:  lkrscss  33895  eqlkr  33896  lkrshp  33902  lkrshp3  33903  lkrshpor  33904  lfl1dim  33918  lfl1dim2N  33919  lkr0f2  33958  lclkrlem1  36303  lclkrlem2j  36313  lclkr  36330  lclkrs  36336  mapd0  36462
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