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Theorem lhpset 30806
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 2809 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 lhpset.h . . 3  |-  H  =  ( LHyp `  K
)
3 fveq2 5541 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 lhpset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2346 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 eqidd 2297 . . . . . 6  |-  ( k  =  K  ->  w  =  w )
7 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
8 lhpset.c . . . . . . 7  |-  C  =  (  <o  `  K )
97, 8syl6eqr 2346 . . . . . 6  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
10 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  ( 1. `  k )  =  ( 1. `  K
) )
11 lhpset.u . . . . . . 7  |-  .1.  =  ( 1. `  K )
1210, 11syl6eqr 2346 . . . . . 6  |-  ( k  =  K  ->  ( 1. `  k )  =  .1.  )
136, 9, 12breq123d 4053 . . . . 5  |-  ( k  =  K  ->  (
w (  <o  `  k
) ( 1. `  k )  <->  w C  .1.  ) )
145, 13rabeqbidv 2796 . . . 4  |-  ( k  =  K  ->  { w  e.  ( Base `  k
)  |  w ( 
<o  `  k ) ( 1. `  k ) }  =  { w  e.  B  |  w C  .1.  } )
15 df-lhyp 30799 . . . 4  |-  LHyp  =  ( k  e.  _V  |->  { w  e.  ( Base `  k )  |  w (  <o  `  k
) ( 1. `  k ) } )
16 fvex 5555 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2366 . . . . 5  |-  B  e. 
_V
1817rabex 4181 . . . 4  |-  { w  e.  B  |  w C  .1.  }  e.  _V
1914, 15, 18fvmpt 5618 . . 3  |-  ( K  e.  _V  ->  ( LHyp `  K )  =  { w  e.  B  |  w C  .1.  }
)
202, 19syl5eq 2340 . 2  |-  ( K  e.  _V  ->  H  =  { w  e.  B  |  w C  .1.  }
)
211, 20syl 15 1  |-  ( K  e.  A  ->  H  =  { w  e.  B  |  w C  .1.  }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   class class class wbr 4039   ` cfv 5271   Basecbs 13164   1.cp1 14160    <o ccvr 30074   LHypclh 30795
This theorem is referenced by:  islhp  30807
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-lhyp 30799
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