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Theorem llnset 29991
Description: The set of lattice lines in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)
Hypotheses
Ref Expression
llnset.b  |-  B  =  ( Base `  K
)
llnset.c  |-  C  =  (  <o  `  K )
llnset.a  |-  A  =  ( Atoms `  K )
llnset.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
llnset  |-  ( K  e.  D  ->  N  =  { x  e.  B  |  E. p  e.  A  p C x } )
Distinct variable groups:    A, p    x, B    x, p, K
Allowed substitution hints:    A( x)    B( p)    C( x, p)    D( x, p)    N( x, p)

Proof of Theorem llnset
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2928 . 2  |-  ( K  e.  D  ->  K  e.  _V )
2 llnset.n . . 3  |-  N  =  ( LLines `  K )
3 fveq2 5691 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 llnset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2458 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5691 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
7 llnset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7syl6eqr 2458 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
9 fveq2 5691 . . . . . . . 8  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
10 llnset.c . . . . . . . 8  |-  C  =  (  <o  `  K )
119, 10syl6eqr 2458 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
1211breqd 4187 . . . . . 6  |-  ( k  =  K  ->  (
p (  <o  `  k
) x  <->  p C x ) )
138, 12rexeqbidv 2881 . . . . 5  |-  ( k  =  K  ->  ( E. p  e.  ( Atoms `  k ) p (  <o  `  k )
x  <->  E. p  e.  A  p C x ) )
145, 13rabeqbidv 2915 . . . 4  |-  ( k  =  K  ->  { x  e.  ( Base `  k
)  |  E. p  e.  ( Atoms `  k )
p (  <o  `  k
) x }  =  { x  e.  B  |  E. p  e.  A  p C x } )
15 df-llines 29984 . . . 4  |-  LLines  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. p  e.  ( Atoms `  k ) p ( 
<o  `  k ) x } )
16 fvex 5705 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2478 . . . . 5  |-  B  e. 
_V
1817rabex 4318 . . . 4  |-  { x  e.  B  |  E. p  e.  A  p C x }  e.  _V
1914, 15, 18fvmpt 5769 . . 3  |-  ( K  e.  _V  ->  ( LLines `
 K )  =  { x  e.  B  |  E. p  e.  A  p C x } )
202, 19syl5eq 2452 . 2  |-  ( K  e.  _V  ->  N  =  { x  e.  B  |  E. p  e.  A  p C x } )
211, 20syl 16 1  |-  ( K  e.  D  ->  N  =  { x  e.  B  |  E. p  e.  A  p C x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   E.wrex 2671   {crab 2674   _Vcvv 2920   class class class wbr 4176   ` cfv 5417   Basecbs 13428    <o ccvr 29749   Atomscatm 29750   LLinesclln 29977
This theorem is referenced by:  islln  29992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-llines 29984
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