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Theorem lkrshp 29222
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 16107 . . . 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 2389 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
74, 5, 6lkrlss 29212 . . 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 29211 . . . . 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 2587 . . 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 2389 . . . 4  |-  ( 1r
`  D )  =  ( 1r `  D
)
1810, 11, 17, 12, 4lfl1 29187 . . 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 16106 . . . . . . . . 9  |-  ( W  e.  LVec  ->  D  e.  DivRing )
2411, 17drngunz 15779 . . . . . . . . 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 2570 . . . . . . 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 29210 . . . . . . . . . 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 2588 . . . . . . 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 2389 . . . . . . 7  |-  ( LSpan `  W )  =  (
LSpan `  W )
3612, 35, 4, 5lkrlsp3 29221 . . . . . 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 2763 . . 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 29096 . . 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 1649    e. wcel 1717    =/= wne 2552   E.wrex 2652    u. cun 3263   {csn 3759    X. cxp 4818   ` cfv 5396   Basecbs 13398  Scalarcsca 13461   0gc0g 13652   1rcur 15591   DivRingcdr 15764   LModclmod 15879   LSubSpclss 15937   LSpanclspn 15976   LVecclvec 16103  LSHypclsh 29092  LFnlclfn 29174  LKerclk 29202
This theorem is referenced by:  lkrshp3  29223  lkrshpor  29224  lshpset2N  29236  lfl1dim  29238  lfl1dim2N  29239  hdmaplkr  32033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-tpos 6417  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-0g 13656  df-mnd 14619  df-submnd 14668  df-grp 14741  df-minusg 14742  df-sbg 14743  df-subg 14870  df-cntz 15045  df-lsm 15199  df-cmn 15343  df-abl 15344  df-mgp 15578  df-rng 15592  df-ur 15594  df-oppr 15657  df-dvdsr 15675  df-unit 15676  df-invr 15706  df-drng 15766  df-lmod 15881  df-lss 15938  df-lsp 15977  df-lvec 16104  df-lshyp 29094  df-lfl 29175  df-lkr 29203
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