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Theorem lkr0f 29355
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 2366 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
3 lkr0f.v . . . . . . 7  |-  V  =  ( Base `  W
)
4 lkr0f.f . . . . . . 7  |-  F  =  (LFnl `  W )
51, 2, 3, 4lflf 29324 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> ( Base `  D
) )
6 ffn 5495 . . . . . 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 29349 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
1211eqeq1d 2374 . . . . 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 5646 . . . . . . 7  |-  ( 0g
`  D )  e. 
_V
159, 14eqeltri 2436 . . . . . 6  |-  .0.  e.  _V
1615fconst2 5848 . . . . 5  |-  ( G : V --> {  .0.  }  <-> 
G  =  ( V  X.  {  .0.  }
) )
17 fconst4 5856 . . . . 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 2366 . . . . . . . . . . 11  |-  (LFnl `  W )  =  (LFnl `  W )
251, 9, 3, 24lfl0f 29330 . . . . . . . . . 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 2440 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  G  e.  (LFnl `  W ) )
281, 9, 24, 10lkrval 29349 . . . . . . . 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 2374 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( ( K `
 G )  =  V  <->  ( `' G " {  .0.  } )  =  V ) )
31 ffn 5495 . . . . . . . . 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 1647    e. wcel 1715   _Vcvv 2873   {csn 3729    X. cxp 4790   `'ccnv 4791   "cima 4795    Fn wfn 5353   -->wf 5354   ` cfv 5358   Basecbs 13356  Scalarcsca 13419   0gc0g 13610   LModclmod 15837  LFnlclfn 29318  LKerclk 29346
This theorem is referenced by:  lkrscss  29359  eqlkr  29360  lkrshp  29366  lkrshp3  29367  lkrshpor  29368  lfl1dim  29382  lfl1dim2N  29383  lkr0f2  29422  lclkrlem1  31767  lclkrlem2j  31777  lclkr  31794  lclkrs  31800  mapd0  31926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-plusg 13429  df-0g 13614  df-mnd 14577  df-grp 14699  df-mgp 15536  df-rng 15550  df-lmod 15839  df-lfl 29319  df-lkr 29347
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