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Theorem lshpset2N 29309
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 29308 . . . . 5  |-  ( ( W  e.  LVec  /\  s  e.  H )  ->  E. g  e.  F  ( K `  g )  =  s )
5 eleq1 2343 . . . . . . . . . . . 12  |-  ( ( K `  g )  =  s  ->  (
( K `  g
)  e.  H  <->  s  e.  H ) )
65biimparc 473 . . . . . . . . . . 11  |-  ( ( s  e.  H  /\  ( K `  g )  =  s )  -> 
( K `  g
)  e.  H )
76adantll 694 . . . . . . . . . 10  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  ( K `  g )  =  s )  ->  ( K `  g )  e.  H
)
87adantlr 695 . . . . . . . . 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 734 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  ->  W  e.  LVec )
13 simplr 731 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  -> 
g  e.  F )
149, 10, 11, 1, 2, 3, 12, 13lkrshp3 29296 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  -> 
( ( K `  g )  e.  H  <->  g  =/=  ( V  X.  {  .0.  } ) ) )
158, 14mpbid 201 . . . . . . . 8  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  -> 
g  =/=  ( V  X.  {  .0.  }
) )
1615ex 423 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  g  e.  F
)  ->  ( ( K `  g )  =  s  ->  g  =/=  ( V  X.  {  .0.  } ) ) )
17 eqimss2 3231 . . . . . . . . 9  |-  ( ( K `  g )  =  s  ->  s  C_  ( K `  g
) )
18 eqimss 3230 . . . . . . . . 9  |-  ( ( K `  g )  =  s  ->  ( K `  g )  C_  s )
1917, 18eqssd 3196 . . . . . . . 8  |-  ( ( K `  g )  =  s  ->  s  =  ( K `  g ) )
2019a1i 10 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  g  e.  F
)  ->  ( ( K `  g )  =  s  ->  s  =  ( K `  g
) ) )
2116, 20jcad 519 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  g  e.  F
)  ->  ( ( K `  g )  =  s  ->  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) ) )
2221reximdva 2655 . . . . 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 14 . . . 4  |-  ( ( W  e.  LVec  /\  s  e.  H )  ->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) )
2423ex 423 . . 3  |-  ( W  e.  LVec  ->  ( s  e.  H  ->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) ) )
259, 10, 11, 1, 2, 3lkrshp 29295 . . . . . . . 8  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  g  =/=  ( V  X.  {  .0.  } ) )  -> 
( K `  g
)  e.  H )
26253adant3r 1179 . . . . . . 7  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) )  ->  ( K `  g )  e.  H )
27 eqid 2283 . . . . . . . . 9  |-  ( LSpan `  W )  =  (
LSpan `  W )
28 eqid 2283 . . . . . . . . 9  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
299, 27, 28, 1islshp 29169 . . . . . . . 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 976 . . . . . . 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 201 . . . . . 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 2343 . . . . . . . . 9  |-  ( s  =  ( K `  g )  ->  (
s  e.  ( LSubSp `  W )  <->  ( K `  g )  e.  (
LSubSp `  W ) ) )
33 neeq1 2454 . . . . . . . . 9  |-  ( s  =  ( K `  g )  ->  (
s  =/=  V  <->  ( K `  g )  =/=  V
) )
34 uneq1 3322 . . . . . . . . . . . 12  |-  ( s  =  ( K `  g )  ->  (
s  u.  { v } )  =  ( ( K `  g
)  u.  { v } ) )
3534fveq2d 5529 . . . . . . . . . . 11  |-  ( s  =  ( K `  g )  ->  (
( LSpan `  W ) `  ( s  u.  {
v } ) )  =  ( ( LSpan `  W ) `  (
( K `  g
)  u.  { v } ) ) )
3635eqeq1d 2291 . . . . . . . . . 10  |-  ( s  =  ( K `  g )  ->  (
( ( LSpan `  W
) `  ( s  u.  { v } ) )  =  V  <->  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) )
3736rexbidv 2564 . . . . . . . . 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 1252 . . . . . . . 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 452 . . . . . . 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 978 . . . . . 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 223 . . . . 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 2669 . . . 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 29169 . . . 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 225 . . 3  |-  ( W  e.  LVec  ->  ( E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) )  ->  s  e.  H ) )
4524, 44impbid 183 . 2  |-  ( W  e.  LVec  ->  ( s  e.  H  <->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) ) )
4645abbi2dv 2398 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   E.wrex 2544    u. cun 3150   {csn 3640    X. cxp 4687   ` cfv 5255   Basecbs 13148  Scalarcsca 13211   0gc0g 13400   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855  LSHypclsh 29165  LFnlclfn 29247  LKerclk 29275
This theorem is referenced by:  islshpkrN  29310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lshyp 29167  df-lfl 29248  df-lkr 29276
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