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Theorem lkr0f 29906
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 2296 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
3 lkr0f.v . . . . . . 7  |-  V  =  ( Base `  W
)
4 lkr0f.f . . . . . . 7  |-  F  =  (LFnl `  W )
51, 2, 3, 4lflf 29875 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> ( Base `  D
) )
6 ffn 5405 . . . . . 6  |-  ( G : V --> ( Base `  D )  ->  G  Fn  V )
75, 6syl 15 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G  Fn  V )
87adantr 451 . . . 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 29900 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
1211eqeq1d 2304 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( K `  G
)  =  V  <->  ( `' G " {  .0.  }
)  =  V ) )
1312biimpa 470 . . . 4  |-  ( ( ( W  e.  LMod  /\  G  e.  F )  /\  ( K `  G )  =  V )  ->  ( `' G " {  .0.  }
)  =  V )
14 fvex 5555 . . . . . . 7  |-  ( 0g
`  D )  e. 
_V
159, 14eqeltri 2366 . . . . . 6  |-  .0.  e.  _V
1615fconst2 5746 . . . . 5  |-  ( G : V --> {  .0.  }  <-> 
G  =  ( V  X.  {  .0.  }
) )
17 fconst4 5752 . . . . 5  |-  ( G : V --> {  .0.  }  <-> 
( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) )
1816, 17bitr3i 242 . . . 4  |-  ( G  =  ( V  X.  {  .0.  } )  <->  ( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) )
198, 13, 18sylanbrc 645 . . 3  |-  ( ( ( W  e.  LMod  /\  G  e.  F )  /\  ( K `  G )  =  V )  ->  G  =  ( V  X.  {  .0.  } ) )
2019ex 423 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( K `  G
)  =  V  ->  G  =  ( V  X.  {  .0.  } ) ) )
2118biimpi 186 . . . . . 6  |-  ( G  =  ( V  X.  {  .0.  } )  -> 
( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) )
2221adantl 452 . . . . 5  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) )
23 simpr 447 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  G  =  ( V  X.  {  .0.  } ) )
24 eqid 2296 . . . . . . . . . . 11  |-  (LFnl `  W )  =  (LFnl `  W )
251, 9, 3, 24lfl0f 29881 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  }
)  e.  (LFnl `  W ) )
2625adantr 451 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( V  X.  {  .0.  } )  e.  (LFnl `  W )
)
2723, 26eqeltrd 2370 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  G  e.  (LFnl `  W ) )
281, 9, 24, 10lkrval 29900 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  (LFnl `  W )
)  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
2927, 28syldan 456 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
3029eqeq1d 2304 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( ( K `
 G )  =  V  <->  ( `' G " {  .0.  } )  =  V ) )
31 ffn 5405 . . . . . . . . 9  |-  ( G : V --> {  .0.  }  ->  G  Fn  V
)
3216, 31sylbir 204 . . . . . . . 8  |-  ( G  =  ( V  X.  {  .0.  } )  ->  G  Fn  V )
3332adantl 452 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  G  Fn  V
)
3433biantrurd 494 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( ( `' G " {  .0.  } )  =  V  <->  ( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) ) )
3530, 34bitrd 244 . . . . 5  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( ( K `
 G )  =  V  <->  ( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) ) )
3622, 35mpbird 223 . . . 4  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( K `  G )  =  V )
3736ex 423 . . 3  |-  ( W  e.  LMod  ->  ( G  =  ( V  X.  {  .0.  } )  -> 
( K `  G
)  =  V ) )
3837adantr 451 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( G  =  ( V  X.  {  .0.  } )  ->  ( K `  G )  =  V ) )
3920, 38impbid 183 1  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( K `  G
)  =  V  <->  G  =  ( V  X.  {  .0.  } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    X. cxp 4703   `'ccnv 4704   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271   Basecbs 13164  Scalarcsca 13227   0gc0g 13416   LModclmod 15643  LFnlclfn 29869  LKerclk 29897
This theorem is referenced by:  lkrscss  29910  eqlkr  29911  lkrshp  29917  lkrshp3  29918  lkrshpor  29919  lfl1dim  29933  lfl1dim2N  29934  lkr0f2  29973  lclkrlem1  32318  lclkrlem2j  32328  lclkr  32345  lclkrs  32351  mapd0  32477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-mgp 15342  df-rng 15356  df-lmod 15645  df-lfl 29870  df-lkr 29898
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