Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhpset Structured version   Unicode version

Theorem lhpset 30854
Description: The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)
Hypotheses
Ref Expression
lhpset.b  |-  B  =  ( Base `  K
)
lhpset.u  |-  .1.  =  ( 1. `  K )
lhpset.c  |-  C  =  (  <o  `  K )
lhpset.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpset  |-  ( K  e.  A  ->  H  =  { w  e.  B  |  w C  .1.  }
)
Distinct variable groups:    w, B    w, C    w, K    w,  .1.
Allowed substitution hints:    A( w)    H( w)

Proof of Theorem lhpset
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2966 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 lhpset.h . . 3  |-  H  =  ( LHyp `  K
)
3 fveq2 5730 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 lhpset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2488 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 eqidd 2439 . . . . . 6  |-  ( k  =  K  ->  w  =  w )
7 fveq2 5730 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
8 lhpset.c . . . . . . 7  |-  C  =  (  <o  `  K )
97, 8syl6eqr 2488 . . . . . 6  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
10 fveq2 5730 . . . . . . 7  |-  ( k  =  K  ->  ( 1. `  k )  =  ( 1. `  K
) )
11 lhpset.u . . . . . . 7  |-  .1.  =  ( 1. `  K )
1210, 11syl6eqr 2488 . . . . . 6  |-  ( k  =  K  ->  ( 1. `  k )  =  .1.  )
136, 9, 12breq123d 4228 . . . . 5  |-  ( k  =  K  ->  (
w (  <o  `  k
) ( 1. `  k )  <->  w C  .1.  ) )
145, 13rabeqbidv 2953 . . . 4  |-  ( k  =  K  ->  { w  e.  ( Base `  k
)  |  w ( 
<o  `  k ) ( 1. `  k ) }  =  { w  e.  B  |  w C  .1.  } )
15 df-lhyp 30847 . . . 4  |-  LHyp  =  ( k  e.  _V  |->  { w  e.  ( Base `  k )  |  w (  <o  `  k
) ( 1. `  k ) } )
16 fvex 5744 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2508 . . . . 5  |-  B  e. 
_V
1817rabex 4356 . . . 4  |-  { w  e.  B  |  w C  .1.  }  e.  _V
1914, 15, 18fvmpt 5808 . . 3  |-  ( K  e.  _V  ->  ( LHyp `  K )  =  { w  e.  B  |  w C  .1.  }
)
202, 19syl5eq 2482 . 2  |-  ( K  e.  _V  ->  H  =  { w  e.  B  |  w C  .1.  }
)
211, 20syl 16 1  |-  ( K  e.  A  ->  H  =  { w  e.  B  |  w C  .1.  }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958   class class class wbr 4214   ` cfv 5456   Basecbs 13471   1.cp1 14469    <o ccvr 30122   LHypclh 30843
This theorem is referenced by:  islhp  30855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-lhyp 30847
  Copyright terms: Public domain W3C validator