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Theorem lshpset2N 29235
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 29234 . . . . 5  |-  ( ( W  e.  LVec  /\  s  e.  H )  ->  E. g  e.  F  ( K `  g )  =  s )
5 eleq1 2448 . . . . . . . . . . . 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 29222 . . . . . . . . 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 3345 . . . . . . . . 9  |-  ( ( K `  g )  =  s  ->  s  C_  ( K `  g
) )
18 eqimss 3344 . . . . . . . . 9  |-  ( ( K `  g )  =  s  ->  ( K `  g )  C_  s )
1917, 18eqssd 3309 . . . . . . . 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 2762 . . . . 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 29221 . . . . . . . 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 2388 . . . . . . . . 9  |-  ( LSpan `  W )  =  (
LSpan `  W )
28 eqid 2388 . . . . . . . . 9  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
299, 27, 28, 1islshp 29095 . . . . . . . 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 2448 . . . . . . . . 9  |-  ( s  =  ( K `  g )  ->  (
s  e.  ( LSubSp `  W )  <->  ( K `  g )  e.  (
LSubSp `  W ) ) )
33 neeq1 2559 . . . . . . . . 9  |-  ( s  =  ( K `  g )  ->  (
s  =/=  V  <->  ( K `  g )  =/=  V
) )
34 uneq1 3438 . . . . . . . . . . . 12  |-  ( s  =  ( K `  g )  ->  (
s  u.  { v } )  =  ( ( K `  g
)  u.  { v } ) )
3534fveq2d 5673 . . . . . . . . . . 11  |-  ( s  =  ( K `  g )  ->  (
( LSpan `  W ) `  ( s  u.  {
v } ) )  =  ( ( LSpan `  W ) `  (
( K `  g
)  u.  { v } ) ) )
3635eqeq1d 2396 . . . . . . . . . 10  |-  ( s  =  ( K `  g )  ->  (
( ( LSpan `  W
) `  ( s  u.  { v } ) )  =  V  <->  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) )
3736rexbidv 2671 . . . . . . . . 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 2776 . . . 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 29095 . . . 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 2503 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 1649    e. wcel 1717   {cab 2374    =/= wne 2551   E.wrex 2651    u. cun 3262   {csn 3758    X. cxp 4817   ` cfv 5395   Basecbs 13397  Scalarcsca 13460   0gc0g 13651   LSubSpclss 15936   LSpanclspn 15975   LVecclvec 16102  LSHypclsh 29091  LFnlclfn 29173  LKerclk 29201
This theorem is referenced by:  islshpkrN  29236
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-tpos 6416  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-0g 13655  df-mnd 14618  df-submnd 14667  df-grp 14740  df-minusg 14741  df-sbg 14742  df-subg 14869  df-cntz 15044  df-lsm 15198  df-cmn 15342  df-abl 15343  df-mgp 15577  df-rng 15591  df-ur 15593  df-oppr 15656  df-dvdsr 15674  df-unit 15675  df-invr 15705  df-drng 15765  df-lmod 15880  df-lss 15937  df-lsp 15976  df-lvec 16103  df-lshyp 29093  df-lfl 29174  df-lkr 29202
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