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

Theorem lhpset 33735
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 3002 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 lhpset.h . . 3  |-  H  =  ( LHyp `  K
)
3 fveq2 5712 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 lhpset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2493 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 eqidd 2444 . . . . . 6  |-  ( k  =  K  ->  w  =  w )
7 fveq2 5712 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
8 lhpset.c . . . . . . 7  |-  C  =  (  <o  `  K )
97, 8syl6eqr 2493 . . . . . 6  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
10 fveq2 5712 . . . . . . 7  |-  ( k  =  K  ->  ( 1. `  k )  =  ( 1. `  K
) )
11 lhpset.u . . . . . . 7  |-  .1.  =  ( 1. `  K )
1210, 11syl6eqr 2493 . . . . . 6  |-  ( k  =  K  ->  ( 1. `  k )  =  .1.  )
136, 9, 12breq123d 4327 . . . . 5  |-  ( k  =  K  ->  (
w (  <o  `  k
) ( 1. `  k )  <->  w C  .1.  ) )
145, 13rabeqbidv 2988 . . . 4  |-  ( k  =  K  ->  { w  e.  ( Base `  k
)  |  w ( 
<o  `  k ) ( 1. `  k ) }  =  { w  e.  B  |  w C  .1.  } )
15 df-lhyp 33728 . . . 4  |-  LHyp  =  ( k  e.  _V  |->  { w  e.  ( Base `  k )  |  w (  <o  `  k
) ( 1. `  k ) } )
16 fvex 5722 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2513 . . . . 5  |-  B  e. 
_V
1817rabex 4464 . . . 4  |-  { w  e.  B  |  w C  .1.  }  e.  _V
1914, 15, 18fvmpt 5795 . . 3  |-  ( K  e.  _V  ->  ( LHyp `  K )  =  { w  e.  B  |  w C  .1.  }
)
202, 19syl5eq 2487 . 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 1369    e. wcel 1756   {crab 2740   _Vcvv 2993   class class class wbr 4313   ` cfv 5439   Basecbs 14195   1.cp1 15229    <o ccvr 33003   LHypclh 33724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-iota 5402  df-fun 5441  df-fv 5447  df-lhyp 33728
This theorem is referenced by:  islhp  33736
  Copyright terms: Public domain W3C validator