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Theorem islshpkrN 29286
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 29284? 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 29285 . . 3  |-  ( W  e.  LVec  ->  H  =  { s  |  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) } )
87eleq2d 2447 . 2  |-  ( W  e.  LVec  ->  ( U  e.  H  <->  U  e.  { s  |  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) } ) )
9 elex 2900 . . . 4  |-  ( U  e.  { s  |  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) }  ->  U  e.  _V )
109adantl 453 . . 3  |-  ( ( W  e.  LVec  /\  U  e.  { s  |  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) } )  ->  U  e.  _V )
11 fvex 5675 . . . . . . 7  |-  ( K `
 g )  e. 
_V
12 eleq1 2440 . . . . . . 7  |-  ( U  =  ( K `  g )  ->  ( U  e.  _V  <->  ( K `  g )  e.  _V ) )
1311, 12mpbiri 225 . . . . . 6  |-  ( U  =  ( K `  g )  ->  U  e.  _V )
1413adantl 453 . . . . 5  |-  ( ( g  =/=  ( V  X.  {  .0.  }
)  /\  U  =  ( K `  g ) )  ->  U  e.  _V )
1514rexlimivw 2762 . . . 4  |-  ( E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  U  =  ( K `  g ) )  ->  U  e.  _V )
1615adantl 453 . . 3  |-  ( ( W  e.  LVec  /\  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  U  =  ( K `  g ) ) )  ->  U  e.  _V )
17 eqeq1 2386 . . . . . 6  |-  ( s  =  U  ->  (
s  =  ( K `
 g )  <->  U  =  ( K `  g ) ) )
1817anbi2d 685 . . . . 5  |-  ( s  =  U  ->  (
( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) )  <->  ( g  =/=  ( V  X.  {  .0.  } )  /\  U  =  ( K `  g ) ) ) )
1918rexbidv 2663 . . . 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 3019 . . 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 869 . 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 245 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2366    =/= wne 2543   E.wrex 2643   _Vcvv 2892   {csn 3750    X. cxp 4809   ` cfv 5387   Basecbs 13389  Scalarcsca 13452   0gc0g 13643   LVecclvec 16094  LSHypclsh 29141  LFnlclfn 29223  LKerclk 29251
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-tpos 6408  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-0g 13647  df-mnd 14610  df-submnd 14659  df-grp 14732  df-minusg 14733  df-sbg 14734  df-subg 14861  df-cntz 15036  df-lsm 15190  df-cmn 15334  df-abl 15335  df-mgp 15569  df-rng 15583  df-ur 15585  df-oppr 15648  df-dvdsr 15666  df-unit 15667  df-invr 15697  df-drng 15757  df-lmod 15872  df-lss 15929  df-lsp 15968  df-lvec 16095  df-lshyp 29143  df-lfl 29224  df-lkr 29252
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