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Theorem lhpset 36116
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 3115 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 lhpset.h . . 3  |-  H  =  ( LHyp `  K
)
3 fveq2 5848 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 lhpset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2513 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 eqidd 2455 . . . . . 6  |-  ( k  =  K  ->  w  =  w )
7 fveq2 5848 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
8 lhpset.c . . . . . . 7  |-  C  =  (  <o  `  K )
97, 8syl6eqr 2513 . . . . . 6  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
10 fveq2 5848 . . . . . . 7  |-  ( k  =  K  ->  ( 1. `  k )  =  ( 1. `  K
) )
11 lhpset.u . . . . . . 7  |-  .1.  =  ( 1. `  K )
1210, 11syl6eqr 2513 . . . . . 6  |-  ( k  =  K  ->  ( 1. `  k )  =  .1.  )
136, 9, 12breq123d 4453 . . . . 5  |-  ( k  =  K  ->  (
w (  <o  `  k
) ( 1. `  k )  <->  w C  .1.  ) )
145, 13rabeqbidv 3101 . . . 4  |-  ( k  =  K  ->  { w  e.  ( Base `  k
)  |  w ( 
<o  `  k ) ( 1. `  k ) }  =  { w  e.  B  |  w C  .1.  } )
15 df-lhyp 36109 . . . 4  |-  LHyp  =  ( k  e.  _V  |->  { w  e.  ( Base `  k )  |  w (  <o  `  k
) ( 1. `  k ) } )
16 fvex 5858 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2538 . . . . 5  |-  B  e. 
_V
1817rabex 4588 . . . 4  |-  { w  e.  B  |  w C  .1.  }  e.  _V
1914, 15, 18fvmpt 5931 . . 3  |-  ( K  e.  _V  ->  ( LHyp `  K )  =  { w  e.  B  |  w C  .1.  }
)
202, 19syl5eq 2507 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106   class class class wbr 4439   ` cfv 5570   Basecbs 14716   1.cp1 15867    <o ccvr 35384   LHypclh 36105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-lhyp 36109
This theorem is referenced by:  islhp  36117
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