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Theorem lkrshp 35246
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 17950 . . . 4  |-  ( W  e.  LVec  ->  W  e. 
LMod )
213ad2ant1 1015 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  W  e.  LMod )
3 simp2 995 . . 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 2454 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
74, 5, 6lkrlss 35236 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  e.  ( LSubSp `  W )
)
82, 3, 7syl2anc 659 . 2  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( K `  G
)  e.  ( LSubSp `  W ) )
9 simp3 996 . . 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 35235 . . . . 5  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( K `  G
)  =  V  <->  G  =  ( V  X.  {  .0.  } ) ) )
142, 3, 13syl2anc 659 . . . 4  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  -> 
( ( K `  G )  =  V  <-> 
G  =  ( V  X.  {  .0.  }
) ) )
1514necon3bid 2712 . . 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 2454 . . . 4  |-  ( 1r
`  D )  =  ( 1r `  D
)
1810, 11, 17, 12, 4lfl1 35211 . . 3  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  ->  E. v  e.  V  ( G `  v )  =  ( 1r `  D ) )
19 simp11 1024 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  W  e.  LVec )
20 simp2 995 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  v  e.  V
)
21 simp12 1025 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  G  e.  F
)
22 simp3 996 . . . . . . . 8  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  ( G `  v )  =  ( 1r `  D ) )
2310lvecdrng 17949 . . . . . . . . 9  |-  ( W  e.  LVec  ->  D  e.  DivRing )
2411, 17drngunz 17609 . . . . . . . . 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 2747 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) )  /\  v  e.  V  /\  ( G `  v
)  =  ( 1r
`  D ) )  ->  ( G `  v )  =/=  .0.  )
27 simpl11 1069 . . . . . . . . . 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 1070 . . . . . . . . . 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 459 . . . . . . . . . 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 35234 . . . . . . . . . 10  |-  ( ( W  e.  LVec  /\  G  e.  F  /\  v  e.  ( K `  G
) )  ->  ( G `  v )  =  .0.  )
3127, 28, 29, 30syl3anc 1226 . . . . . . . . 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 432 . . . . . . . 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 2664 . . . . . . 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 2454 . . . . . . 7  |-  ( LSpan `  W )  =  (
LSpan `  W )
3612, 35, 4, 5lkrlsp3 35245 . . . . . 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 1231 . . . . 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 1196 . . . 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 2929 . . 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 35120 . . 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 1015 . 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 1177 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805    u. cun 3459   {csn 4016    X. cxp 4986   ` cfv 5570   Basecbs 14719  Scalarcsca 14790   0gc0g 14932   1rcur 17351   DivRingcdr 17594   LModclmod 17710   LSubSpclss 17776   LSpanclspn 17815   LVecclvec 17946  LSHypclsh 35116  LFnlclfn 35198  LKerclk 35226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-grp 16259  df-minusg 16260  df-sbg 16261  df-subg 16400  df-cntz 16557  df-lsm 16858  df-cmn 17002  df-abl 17003  df-mgp 17340  df-ur 17352  df-ring 17398  df-oppr 17470  df-dvdsr 17488  df-unit 17489  df-invr 17519  df-drng 17596  df-lmod 17712  df-lss 17777  df-lsp 17816  df-lvec 17947  df-lshyp 35118  df-lfl 35199  df-lkr 35227
This theorem is referenced by:  lkrshp3  35247  lkrshpor  35248  lshpset2N  35260  lfl1dim  35262  lfl1dim2N  35263  hdmaplkr  38059
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