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Theorem lshpset2N 29854
Description: The set of all hyperplanes of a left module or left vector space equals the set of all kernels of nonzero functionals. (Contributed by NM, 17-Jul-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lshpset2.v  |-  V  =  ( Base `  W
)
lshpset2.d  |-  D  =  (Scalar `  W )
lshpset2.z  |-  .0.  =  ( 0g `  D )
lshpset2.h  |-  H  =  (LSHyp `  W )
lshpset2.f  |-  F  =  (LFnl `  W )
lshpset2.k  |-  K  =  (LKer `  W )
Assertion
Ref Expression
lshpset2N  |-  ( W  e.  LVec  ->  H  =  { s  |  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) } )
Distinct variable groups:    g, F    g, s, H    g, K    g, V    g, W, s
Allowed substitution hints:    D( g, s)    F( s)    K( s)    V( s)    .0. ( g, s)

Proof of Theorem lshpset2N
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lshpset2.h . . . . . 6  |-  H  =  (LSHyp `  W )
2 lshpset2.f . . . . . 6  |-  F  =  (LFnl `  W )
3 lshpset2.k . . . . . 6  |-  K  =  (LKer `  W )
41, 2, 3lshpkrex 29853 . . . . 5  |-  ( ( W  e.  LVec  /\  s  e.  H )  ->  E. g  e.  F  ( K `  g )  =  s )
5 eleq1 2495 . . . . . . . . . . . 12  |-  ( ( K `  g )  =  s  ->  (
( K `  g
)  e.  H  <->  s  e.  H ) )
65biimparc 474 . . . . . . . . . . 11  |-  ( ( s  e.  H  /\  ( K `  g )  =  s )  -> 
( K `  g
)  e.  H )
76adantll 695 . . . . . . . . . 10  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  ( K `  g )  =  s )  ->  ( K `  g )  e.  H
)
87adantlr 696 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  -> 
( K `  g
)  e.  H )
9 lshpset2.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
10 lshpset2.d . . . . . . . . . 10  |-  D  =  (Scalar `  W )
11 lshpset2.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  D )
12 simplll 735 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  ->  W  e.  LVec )
13 simplr 732 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  -> 
g  e.  F )
149, 10, 11, 1, 2, 3, 12, 13lkrshp3 29841 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  -> 
( ( K `  g )  e.  H  <->  g  =/=  ( V  X.  {  .0.  } ) ) )
158, 14mpbid 202 . . . . . . . 8  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  -> 
g  =/=  ( V  X.  {  .0.  }
) )
1615ex 424 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  g  e.  F
)  ->  ( ( K `  g )  =  s  ->  g  =/=  ( V  X.  {  .0.  } ) ) )
17 eqimss2 3393 . . . . . . . . 9  |-  ( ( K `  g )  =  s  ->  s  C_  ( K `  g
) )
18 eqimss 3392 . . . . . . . . 9  |-  ( ( K `  g )  =  s  ->  ( K `  g )  C_  s )
1917, 18eqssd 3357 . . . . . . . 8  |-  ( ( K `  g )  =  s  ->  s  =  ( K `  g ) )
2019a1i 11 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  g  e.  F
)  ->  ( ( K `  g )  =  s  ->  s  =  ( K `  g
) ) )
2116, 20jcad 520 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  g  e.  F
)  ->  ( ( K `  g )  =  s  ->  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) ) )
2221reximdva 2810 . . . . 5  |-  ( ( W  e.  LVec  /\  s  e.  H )  ->  ( E. g  e.  F  ( K `  g )  =  s  ->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) ) )
234, 22mpd 15 . . . 4  |-  ( ( W  e.  LVec  /\  s  e.  H )  ->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) )
2423ex 424 . . 3  |-  ( W  e.  LVec  ->  ( s  e.  H  ->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) ) )
259, 10, 11, 1, 2, 3lkrshp 29840 . . . . . . . 8  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  g  =/=  ( V  X.  {  .0.  } ) )  -> 
( K `  g
)  e.  H )
26253adant3r 1181 . . . . . . 7  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) )  ->  ( K `  g )  e.  H )
27 eqid 2435 . . . . . . . . 9  |-  ( LSpan `  W )  =  (
LSpan `  W )
28 eqid 2435 . . . . . . . . 9  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
299, 27, 28, 1islshp 29714 . . . . . . . 8  |-  ( 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 ) ) )
30293ad2ant1 978 . . . . . . 7  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) )  ->  (
( K `  g
)  e.  H  <->  ( ( K `  g )  e.  ( LSubSp `  W )  /\  ( K `  g
)  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) ) )
3126, 30mpbid 202 . . . . . 6  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) )  ->  (
( K `  g
)  e.  ( LSubSp `  W )  /\  ( K `  g )  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) )
32 eleq1 2495 . . . . . . . . 9  |-  ( s  =  ( K `  g )  ->  (
s  e.  ( LSubSp `  W )  <->  ( K `  g )  e.  (
LSubSp `  W ) ) )
33 neeq1 2606 . . . . . . . . 9  |-  ( s  =  ( K `  g )  ->  (
s  =/=  V  <->  ( K `  g )  =/=  V
) )
34 uneq1 3486 . . . . . . . . . . . 12  |-  ( s  =  ( K `  g )  ->  (
s  u.  { v } )  =  ( ( K `  g
)  u.  { v } ) )
3534fveq2d 5724 . . . . . . . . . . 11  |-  ( s  =  ( K `  g )  ->  (
( LSpan `  W ) `  ( s  u.  {
v } ) )  =  ( ( LSpan `  W ) `  (
( K `  g
)  u.  { v } ) ) )
3635eqeq1d 2443 . . . . . . . . . 10  |-  ( s  =  ( K `  g )  ->  (
( ( LSpan `  W
) `  ( s  u.  { v } ) )  =  V  <->  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) )
3736rexbidv 2718 . . . . . . . . 9  |-  ( s  =  ( K `  g )  ->  ( E. v  e.  V  ( ( LSpan `  W
) `  ( s  u.  { v } ) )  =  V  <->  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) )
3832, 33, 373anbi123d 1254 . . . . . . . 8  |-  ( s  =  ( K `  g )  ->  (
( s  e.  (
LSubSp `  W )  /\  s  =/=  V  /\  E. v  e.  V  (
( LSpan `  W ) `  ( s  u.  {
v } ) )  =  V )  <->  ( ( K `  g )  e.  ( LSubSp `  W )  /\  ( K `  g
)  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) ) )
3938adantl 453 . . . . . . 7  |-  ( ( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) )  ->  ( (
s  e.  ( LSubSp `  W )  /\  s  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( s  u.  {
v } ) )  =  V )  <->  ( ( K `  g )  e.  ( LSubSp `  W )  /\  ( K `  g
)  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) ) )
40393ad2ant3 980 . . . . . 6  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) )  ->  (
( s  e.  (
LSubSp `  W )  /\  s  =/=  V  /\  E. v  e.  V  (
( LSpan `  W ) `  ( s  u.  {
v } ) )  =  V )  <->  ( ( K `  g )  e.  ( LSubSp `  W )  /\  ( K `  g
)  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) ) )
4131, 40mpbird 224 . . . . 5  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) )  ->  (
s  e.  ( LSubSp `  W )  /\  s  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( s  u.  {
v } ) )  =  V ) )
4241rexlimdv3a 2824 . . . 4  |-  ( W  e.  LVec  ->  ( E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) )  ->  ( s  e.  ( LSubSp `  W )  /\  s  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W
) `  ( s  u.  { v } ) )  =  V ) ) )
439, 27, 28, 1islshp 29714 . . . 4  |-  ( W  e.  LVec  ->  ( s  e.  H  <->  ( s  e.  ( LSubSp `  W )  /\  s  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W
) `  ( s  u.  { v } ) )  =  V ) ) )
4442, 43sylibrd 226 . . 3  |-  ( W  e.  LVec  ->  ( E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) )  ->  s  e.  H ) )
4524, 44impbid 184 . 2  |-  ( W  e.  LVec  ->  ( s  e.  H  <->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) ) )
4645abbi2dv 2550 1  |-  ( W  e.  LVec  ->  H  =  { s  |  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   E.wrex 2698    u. cun 3310   {csn 3806    X. cxp 4868   ` cfv 5446   Basecbs 13461  Scalarcsca 13524   0gc0g 13715   LSubSpclss 16000   LSpanclspn 16039   LVecclvec 16166  LSHypclsh 29710  LFnlclfn 29792  LKerclk 29820
This theorem is referenced by:  islshpkrN  29855
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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