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Theorem lkrshp 29840
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 16170 . . . 4  |-  ( W  e.  LVec  ->  W  e. 
LMod )
213ad2ant1 978 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  W  e.  LMod )
3 simp2 958 . . 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 2435 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
74, 5, 6lkrlss 29830 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  e.  ( LSubSp `  W )
)
82, 3, 7syl2anc 643 . 2  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( K `  G
)  e.  ( LSubSp `  W ) )
9 simp3 959 . . 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 29829 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( K `  G
)  =  V  <->  G  =  ( V  X.  {  .0.  } ) ) )
142, 3, 13syl2anc 643 . . . 4  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( ( K `  G )  =  V  <-> 
G  =  ( V  X.  {  .0.  }
) ) )
1514necon3bid 2633 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( ( K `  G )  =/=  V  <->  G  =/=  ( V  X.  {  .0.  } ) ) )
169, 15mpbird 224 . 2  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( K `  G
)  =/=  V )
17 eqid 2435 . . . 4  |-  ( 1r
`  D )  =  ( 1r `  D
)
1810, 11, 17, 12, 4lfl1 29805 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  E. v  e.  V  ( G `  v )  =  ( 1r `  D ) )
19 simp11 987 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  W  e.  LVec )
20 simp2 958 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  v  e.  V
)
21 simp12 988 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  G  e.  F
)
22 simp3 959 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  ( G `  v )  =  ( 1r `  D ) )
2310lvecdrng 16169 . . . . . . . . 9  |-  ( W  e.  LVec  ->  D  e.  DivRing )
2411, 17drngunz 15842 . . . . . . . . 9  |-  ( D  e.  DivRing  ->  ( 1r `  D )  =/=  .0.  )
2519, 23, 243syl 19 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  ( 1r `  D )  =/=  .0.  )
2622, 25eqnetrd 2616 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  ( G `  v )  =/=  .0.  )
27 simpl11 1032 . . . . . . . . . 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 1033 . . . . . . . . . 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 448 . . . . . . . . . 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 29828 . . . . . . . . . 10  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  v  e.  ( K `  G
) )  ->  ( G `  v )  =  .0.  )
3127, 28, 29, 30syl3anc 1184 . . . . . . . . 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 424 . . . . . . . 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 2634 . . . . . . 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 2435 . . . . . . 7  |-  ( LSpan `  W )  =  (
LSpan `  W )
3612, 35, 4, 5lkrlsp3 29839 . . . . . 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 1189 . . . . 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 1155 . . . 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 2810 . . 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 29714 . . 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 978 . 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 1137 1  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( K `  G
)  e.  H )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    u. cun 3310   {csn 3806    X. cxp 4868   ` cfv 5446   Basecbs 13461  Scalarcsca 13524   0gc0g 13715   1rcur 15654   DivRingcdr 15827   LModclmod 15942   LSubSpclss 16000   LSpanclspn 16039   LVecclvec 16166  LSHypclsh 29710  LFnlclfn 29792  LKerclk 29820
This theorem is referenced by:  lkrshp3  29841  lkrshpor  29842  lshpset2N  29854  lfl1dim  29856  lfl1dim2N  29857  hdmaplkr  32651
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-0g 13719  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lshyp 29712  df-lfl 29793  df-lkr 29821
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