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Theorem islshpkrN 32138
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 32136? 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 32137 . . 3  |-  ( W  e.  LVec  ->  H  =  { s  |  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) } )
87eleq2d 2472 . 2  |-  ( W  e.  LVec  ->  ( U  e.  H  <->  U  e.  { s  |  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) } ) )
9 elex 3068 . . . 4  |-  ( U  e.  { s  |  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) }  ->  U  e.  _V )
109adantl 464 . . 3  |-  ( ( W  e.  LVec  /\  U  e.  { s  |  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) } )  ->  U  e.  _V )
11 fvex 5859 . . . . . . 7  |-  ( K `
 g )  e. 
_V
12 eleq1 2474 . . . . . . 7  |-  ( U  =  ( K `  g )  ->  ( U  e.  _V  <->  ( K `  g )  e.  _V ) )
1311, 12mpbiri 233 . . . . . 6  |-  ( U  =  ( K `  g )  ->  U  e.  _V )
1413adantl 464 . . . . 5  |-  ( ( g  =/=  ( V  X.  {  .0.  }
)  /\  U  =  ( K `  g ) )  ->  U  e.  _V )
1514rexlimivw 2893 . . . 4  |-  ( E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  U  =  ( K `  g ) )  ->  U  e.  _V )
1615adantl 464 . . 3  |-  ( ( W  e.  LVec  /\  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  U  =  ( K `  g ) ) )  ->  U  e.  _V )
17 eqeq1 2406 . . . . . 6  |-  ( s  =  U  ->  (
s  =  ( K `
 g )  <->  U  =  ( K `  g ) ) )
1817anbi2d 702 . . . . 5  |-  ( s  =  U  ->  (
( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) )  <->  ( g  =/=  ( V  X.  {  .0.  } )  /\  U  =  ( K `  g ) ) ) )
1918rexbidv 2918 . . . 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 3197 . . 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 901 . 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 253 1  |-  ( W  e.  LVec  ->  ( U  e.  H  <->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  U  =  ( K `  g ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387    =/= wne 2598   E.wrex 2755   _Vcvv 3059   {csn 3972    X. cxp 4821   ` cfv 5569   Basecbs 14841  Scalarcsca 14912   0gc0g 15054   LVecclvec 18068  LSHypclsh 31993  LFnlclfn 32075  LKerclk 32103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-tpos 6958  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-subg 16522  df-cntz 16679  df-lsm 16980  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-oppr 17592  df-dvdsr 17610  df-unit 17611  df-invr 17641  df-drng 17718  df-lmod 17834  df-lss 17899  df-lsp 17938  df-lvec 18069  df-lshyp 31995  df-lfl 32076  df-lkr 32104
This theorem is referenced by: (None)
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