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Theorem islshpkrN 29932
Description: The predicate "is a hyperplane" (of a left module or left vector space). TODO: should it be 
U  =  ( K `
 g ) or  ( K `  g )  =  U as in lshpkrex 29930? Both standards seem to be used randomly throughout set.mm; we should decide on a preferred one. (Contributed by NM, 7-Oct-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
islshpkrN  |-  ( W  e.  LVec  ->  ( U  e.  H  <->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  U  =  ( K `  g ) ) ) )
Distinct variable groups:    g, F    g, H    g, K    g, V    g, W    U, g
Allowed substitution hints:    D( g)    .0. ( g)

Proof of Theorem islshpkrN
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 lshpset2.v . . . 4  |-  V  =  ( Base `  W
)
2 lshpset2.d . . . 4  |-  D  =  (Scalar `  W )
3 lshpset2.z . . . 4  |-  .0.  =  ( 0g `  D )
4 lshpset2.h . . . 4  |-  H  =  (LSHyp `  W )
5 lshpset2.f . . . 4  |-  F  =  (LFnl `  W )
6 lshpset2.k . . . 4  |-  K  =  (LKer `  W )
71, 2, 3, 4, 5, 6lshpset2N 29931 . . 3  |-  ( W  e.  LVec  ->  H  =  { s  |  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) } )
87eleq2d 2363 . 2  |-  ( W  e.  LVec  ->  ( U  e.  H  <->  U  e.  { s  |  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) } ) )
9 elex 2809 . . . 4  |-  ( U  e.  { s  |  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) }  ->  U  e.  _V )
109adantl 452 . . 3  |-  ( ( W  e.  LVec  /\  U  e.  { s  |  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) } )  ->  U  e.  _V )
11 fvex 5555 . . . . . . 7  |-  ( K `
 g )  e. 
_V
12 eleq1 2356 . . . . . . 7  |-  ( U  =  ( K `  g )  ->  ( U  e.  _V  <->  ( K `  g )  e.  _V ) )
1311, 12mpbiri 224 . . . . . 6  |-  ( U  =  ( K `  g )  ->  U  e.  _V )
1413adantl 452 . . . . 5  |-  ( ( g  =/=  ( V  X.  {  .0.  }
)  /\  U  =  ( K `  g ) )  ->  U  e.  _V )
1514rexlimivw 2676 . . . 4  |-  ( E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  U  =  ( K `  g ) )  ->  U  e.  _V )
1615adantl 452 . . 3  |-  ( ( W  e.  LVec  /\  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  U  =  ( K `  g ) ) )  ->  U  e.  _V )
17 eqeq1 2302 . . . . . 6  |-  ( s  =  U  ->  (
s  =  ( K `
 g )  <->  U  =  ( K `  g ) ) )
1817anbi2d 684 . . . . 5  |-  ( s  =  U  ->  (
( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) )  <->  ( g  =/=  ( V  X.  {  .0.  } )  /\  U  =  ( K `  g ) ) ) )
1918rexbidv 2577 . . . 4  |-  ( s  =  U  ->  ( E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) )  <->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  U  =  ( K `  g ) ) ) )
2019elabg 2928 . . 3  |-  ( U  e.  _V  ->  ( U  e.  { s  |  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) }  <->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  U  =  ( K `  g ) ) ) )
2110, 16, 20pm5.21nd 868 . 2  |-  ( W  e.  LVec  ->  ( U  e.  { s  |  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) }  <->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  U  =  ( K `  g ) ) ) )
228, 21bitrd 244 1  |-  ( W  e.  LVec  ->  ( U  e.  H  <->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  U  =  ( K `  g ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   E.wrex 2557   _Vcvv 2801   {csn 3653    X. cxp 4703   ` cfv 5271   Basecbs 13164  Scalarcsca 13227   0gc0g 13416   LVecclvec 15871  LSHypclsh 29787  LFnlclfn 29869  LKerclk 29897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lshyp 29789  df-lfl 29870  df-lkr 29898
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