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Theorem lkrshp 33053
Description: The kernel of a nonzero functional is a hyperplane. (Contributed by NM, 29-Jun-2014.)
Hypotheses
Ref Expression
lkrshp.v  |-  V  =  ( Base `  W
)
lkrshp.d  |-  D  =  (Scalar `  W )
lkrshp.z  |-  .0.  =  ( 0g `  D )
lkrshp.h  |-  H  =  (LSHyp `  W )
lkrshp.f  |-  F  =  (LFnl `  W )
lkrshp.k  |-  K  =  (LKer `  W )
Assertion
Ref Expression
lkrshp  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( K `  G
)  e.  H )

Proof of Theorem lkrshp
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lveclmod 17290 . . . 4  |-  ( W  e.  LVec  ->  W  e. 
LMod )
213ad2ant1 1009 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  W  e.  LMod )
3 simp2 989 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  G  e.  F )
4 lkrshp.f . . . 4  |-  F  =  (LFnl `  W )
5 lkrshp.k . . . 4  |-  K  =  (LKer `  W )
6 eqid 2451 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
74, 5, 6lkrlss 33043 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  e.  ( LSubSp `  W )
)
82, 3, 7syl2anc 661 . 2  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( K `  G
)  e.  ( LSubSp `  W ) )
9 simp3 990 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  G  =/=  ( V  X.  {  .0.  } ) )
10 lkrshp.d . . . . . 6  |-  D  =  (Scalar `  W )
11 lkrshp.z . . . . . 6  |-  .0.  =  ( 0g `  D )
12 lkrshp.v . . . . . 6  |-  V  =  ( Base `  W
)
1310, 11, 12, 4, 5lkr0f 33042 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( K `  G
)  =  V  <->  G  =  ( V  X.  {  .0.  } ) ) )
142, 3, 13syl2anc 661 . . . 4  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( ( K `  G )  =  V  <-> 
G  =  ( V  X.  {  .0.  }
) ) )
1514necon3bid 2704 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( ( K `  G )  =/=  V  <->  G  =/=  ( V  X.  {  .0.  } ) ) )
169, 15mpbird 232 . 2  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( K `  G
)  =/=  V )
17 eqid 2451 . . . 4  |-  ( 1r
`  D )  =  ( 1r `  D
)
1810, 11, 17, 12, 4lfl1 33018 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  E. v  e.  V  ( G `  v )  =  ( 1r `  D ) )
19 simp11 1018 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  W  e.  LVec )
20 simp2 989 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  v  e.  V
)
21 simp12 1019 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  G  e.  F
)
22 simp3 990 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  ( G `  v )  =  ( 1r `  D ) )
2310lvecdrng 17289 . . . . . . . . 9  |-  ( W  e.  LVec  ->  D  e.  DivRing )
2411, 17drngunz 16950 . . . . . . . . 9  |-  ( D  e.  DivRing  ->  ( 1r `  D )  =/=  .0.  )
2519, 23, 243syl 20 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  ( 1r `  D )  =/=  .0.  )
2622, 25eqnetrd 2739 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  ( G `  v )  =/=  .0.  )
27 simpl11 1063 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  }
) )  /\  v  e.  V  /\  ( G `  v )  =  ( 1r `  D ) )  /\  v  e.  ( K `  G ) )  ->  W  e.  LVec )
28 simpl12 1064 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  }
) )  /\  v  e.  V  /\  ( G `  v )  =  ( 1r `  D ) )  /\  v  e.  ( K `  G ) )  ->  G  e.  F )
29 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  }
) )  /\  v  e.  V  /\  ( G `  v )  =  ( 1r `  D ) )  /\  v  e.  ( K `  G ) )  -> 
v  e.  ( K `
 G ) )
3010, 11, 4, 5lkrf0 33041 . . . . . . . . . 10  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  v  e.  ( K `  G
) )  ->  ( G `  v )  =  .0.  )
3127, 28, 29, 30syl3anc 1219 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  }
) )  /\  v  e.  V  /\  ( G `  v )  =  ( 1r `  D ) )  /\  v  e.  ( K `  G ) )  -> 
( G `  v
)  =  .0.  )
3231ex 434 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  ( v  e.  ( K `  G
)  ->  ( G `  v )  =  .0.  ) )
3332necon3ad 2656 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  ( ( G `
 v )  =/= 
.0.  ->  -.  v  e.  ( K `  G ) ) )
3426, 33mpd 15 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  -.  v  e.  ( K `  G ) )
35 eqid 2451 . . . . . . 7  |-  ( LSpan `  W )  =  (
LSpan `  W )
3612, 35, 4, 5lkrlsp3 33052 . . . . . 6  |-  ( ( W  e.  LVec  /\  (
v  e.  V  /\  G  e.  F )  /\  -.  v  e.  ( K `  G ) )  ->  ( ( LSpan `  W ) `  ( ( K `  G )  u.  {
v } ) )  =  V )
3719, 20, 21, 34, 36syl121anc 1224 . . . . 5  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  ( ( LSpan `  W ) `  (
( K `  G
)  u.  { v } ) )  =  V )
38373expia 1190 . . . 4  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V
)  ->  ( ( G `  v )  =  ( 1r `  D )  ->  (
( LSpan `  W ) `  ( ( K `  G )  u.  {
v } ) )  =  V ) )
3938reximdva 2921 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( E. v  e.  V  ( G `  v )  =  ( 1r `  D )  ->  E. v  e.  V  ( ( LSpan `  W
) `  ( ( K `  G )  u.  { v } ) )  =  V ) )
4018, 39mpd 15 . 2  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  E. v  e.  V  ( ( LSpan `  W
) `  ( ( K `  G )  u.  { v } ) )  =  V )
41 lkrshp.h . . . 4  |-  H  =  (LSHyp `  W )
4212, 35, 6, 41islshp 32927 . . 3  |-  ( W  e.  LVec  ->  ( ( K `  G )  e.  H  <->  ( ( K `  G )  e.  ( LSubSp `  W )  /\  ( K `  G
)  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  G )  u.  {
v } ) )  =  V ) ) )
43423ad2ant1 1009 . 2  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( ( K `  G )  e.  H  <->  ( ( K `  G
)  e.  ( LSubSp `  W )  /\  ( K `  G )  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  G )  u.  {
v } ) )  =  V ) ) )
448, 16, 40, 43mpbir3and 1171 1  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( K `  G
)  e.  H )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2642   E.wrex 2794    u. cun 3421   {csn 3972    X. cxp 4933   ` cfv 5513   Basecbs 14273  Scalarcsca 14340   0gc0g 14477   1rcur 16705   DivRingcdr 16935   LModclmod 17051   LSubSpclss 17116   LSpanclspn 17155   LVecclvec 17286  LSHypclsh 32923  LFnlclfn 33005  LKerclk 33033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-tpos 6842  df-recs 6929  df-rdg 6963  df-er 7198  df-map 7313  df-en 7408  df-dom 7409  df-sdom 7410  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-2 10478  df-3 10479  df-ndx 14276  df-slot 14277  df-base 14278  df-sets 14279  df-ress 14280  df-plusg 14350  df-mulr 14351  df-0g 14479  df-mnd 15514  df-submnd 15564  df-grp 15644  df-minusg 15645  df-sbg 15646  df-subg 15777  df-cntz 15934  df-lsm 16236  df-cmn 16380  df-abl 16381  df-mgp 16694  df-ur 16706  df-rng 16750  df-oppr 16818  df-dvdsr 16836  df-unit 16837  df-invr 16867  df-drng 16937  df-lmod 17053  df-lss 17117  df-lsp 17156  df-lvec 17287  df-lshyp 32925  df-lfl 33006  df-lkr 33034
This theorem is referenced by:  lkrshp3  33054  lkrshpor  33055  lshpset2N  33067  lfl1dim  33069  lfl1dim2N  33070  hdmaplkr  35864
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