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Theorem lkr0f 29623
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 2430 . . . . . . 7  |-  ( Base `  D )  =  (
Base `  D )
3 lkr0f.v . . . . . . 7  |-  V  =  ( Base `  W
)
4 lkr0f.f . . . . . . 7  |-  F  =  (LFnl `  W )
51, 2, 3, 4lflf 29592 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G : V --> ( Base `  D
) )
6 ffn 5577 . . . . . 6  |-  ( G : V --> ( Base `  D )  ->  G  Fn  V )
75, 6syl 16 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  G  Fn  V )
87adantr 452 . . . 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 29617 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
1211eqeq1d 2438 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( K `  G
)  =  V  <->  ( `' G " {  .0.  }
)  =  V ) )
1312biimpa 471 . . . 4  |-  ( ( ( W  e.  LMod  /\  G  e.  F )  /\  ( K `  G )  =  V )  ->  ( `' G " {  .0.  }
)  =  V )
14 fvex 5728 . . . . . . 7  |-  ( 0g
`  D )  e. 
_V
159, 14eqeltri 2500 . . . . . 6  |-  .0.  e.  _V
1615fconst2 5934 . . . . 5  |-  ( G : V --> {  .0.  }  <-> 
G  =  ( V  X.  {  .0.  }
) )
17 fconst4 5942 . . . . 5  |-  ( G : V --> {  .0.  }  <-> 
( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) )
1816, 17bitr3i 243 . . . 4  |-  ( G  =  ( V  X.  {  .0.  } )  <->  ( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) )
198, 13, 18sylanbrc 646 . . 3  |-  ( ( ( W  e.  LMod  /\  G  e.  F )  /\  ( K `  G )  =  V )  ->  G  =  ( V  X.  {  .0.  } ) )
2019ex 424 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( K `  G
)  =  V  ->  G  =  ( V  X.  {  .0.  } ) ) )
2118biimpi 187 . . . . . 6  |-  ( G  =  ( V  X.  {  .0.  } )  -> 
( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) )
2221adantl 453 . . . . 5  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) )
23 simpr 448 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  G  =  ( V  X.  {  .0.  } ) )
24 eqid 2430 . . . . . . . . . . 11  |-  (LFnl `  W )  =  (LFnl `  W )
251, 9, 3, 24lfl0f 29598 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  }
)  e.  (LFnl `  W ) )
2625adantr 452 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( V  X.  {  .0.  } )  e.  (LFnl `  W )
)
2723, 26eqeltrd 2504 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  G  e.  (LFnl `  W ) )
281, 9, 24, 10lkrval 29617 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  (LFnl `  W )
)  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
2927, 28syldan 457 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
3029eqeq1d 2438 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( ( K `
 G )  =  V  <->  ( `' G " {  .0.  } )  =  V ) )
31 ffn 5577 . . . . . . . . 9  |-  ( G : V --> {  .0.  }  ->  G  Fn  V
)
3216, 31sylbir 205 . . . . . . . 8  |-  ( G  =  ( V  X.  {  .0.  } )  ->  G  Fn  V )
3332adantl 453 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  G  Fn  V
)
3433biantrurd 495 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( ( `' G " {  .0.  } )  =  V  <->  ( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) ) )
3530, 34bitrd 245 . . . . 5  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( ( K `
 G )  =  V  <->  ( G  Fn  V  /\  ( `' G " {  .0.  } )  =  V ) ) )
3622, 35mpbird 224 . . . 4  |-  ( ( W  e.  LMod  /\  G  =  ( V  X.  {  .0.  } ) )  ->  ( K `  G )  =  V )
3736ex 424 . . 3  |-  ( W  e.  LMod  ->  ( G  =  ( V  X.  {  .0.  } )  -> 
( K `  G
)  =  V ) )
3837adantr 452 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( G  =  ( V  X.  {  .0.  } )  ->  ( K `  G )  =  V ) )
3920, 38impbid 184 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2943   {csn 3801    X. cxp 4862   `'ccnv 4863   "cima 4867    Fn wfn 5435   -->wf 5436   ` cfv 5440   Basecbs 13452  Scalarcsca 13515   0gc0g 13706   LModclmod 15933  LFnlclfn 29586  LKerclk 29614
This theorem is referenced by:  lkrscss  29627  eqlkr  29628  lkrshp  29634  lkrshp3  29635  lkrshpor  29636  lfl1dim  29650  lfl1dim2N  29651  lkr0f2  29690  lclkrlem1  32035  lclkrlem2j  32045  lclkr  32062  lclkrs  32068  mapd0  32194
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-riota 6535  df-recs 6619  df-rdg 6654  df-er 6891  df-map 7006  df-en 7096  df-dom 7097  df-sdom 7098  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-nn 9985  df-2 10042  df-ndx 13455  df-slot 13456  df-base 13457  df-sets 13458  df-plusg 13525  df-0g 13710  df-mnd 14673  df-grp 14795  df-mgp 15632  df-rng 15646  df-lmod 15935  df-lfl 29587  df-lkr 29615
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