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Theorem islshpkrN 29855
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 29853? 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 29854 . . 3  |-  ( W  e.  LVec  ->  H  =  { s  |  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) } )
87eleq2d 2502 . 2  |-  ( W  e.  LVec  ->  ( U  e.  H  <->  U  e.  { s  |  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) } ) )
9 elex 2956 . . . 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 5734 . . . . . . 7  |-  ( K `
 g )  e. 
_V
12 eleq1 2495 . . . . . . 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 2818 . . . 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 2441 . . . . . 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 2718 . . . 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 3075 . . 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 1652    e. wcel 1725   {cab 2421    =/= wne 2598   E.wrex 2698   _Vcvv 2948   {csn 3806    X. cxp 4868   ` cfv 5446   Basecbs 13461  Scalarcsca 13524   0gc0g 13715   LVecclvec 16166  LSHypclsh 29710  LFnlclfn 29792  LKerclk 29820
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-0g 13719  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lshyp 29712  df-lfl 29793  df-lkr 29821
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