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Theorem llnset 29765
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 2881 . 2  |-  ( K  e.  D  ->  K  e.  _V )
2 llnset.n . . 3  |-  N  =  ( LLines `  K )
3 fveq2 5632 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
4 llnset.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2416 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
6 fveq2 5632 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
7 llnset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7syl6eqr 2416 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
9 fveq2 5632 . . . . . . . 8  |-  ( k  =  K  ->  (  <o  `  k )  =  (  <o  `  K )
)
10 llnset.c . . . . . . . 8  |-  C  =  (  <o  `  K )
119, 10syl6eqr 2416 . . . . . . 7  |-  ( k  =  K  ->  (  <o  `  k )  =  C )
1211breqd 4136 . . . . . 6  |-  ( k  =  K  ->  (
p (  <o  `  k
) x  <->  p C x ) )
138, 12rexeqbidv 2834 . . . . 5  |-  ( k  =  K  ->  ( E. p  e.  ( Atoms `  k ) p (  <o  `  k )
x  <->  E. p  e.  A  p C x ) )
145, 13rabeqbidv 2868 . . . 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 29758 . . . 4  |-  LLines  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. p  e.  ( Atoms `  k ) p ( 
<o  `  k ) x } )
16 fvex 5646 . . . . . 6  |-  ( Base `  K )  e.  _V
174, 16eqeltri 2436 . . . . 5  |-  B  e. 
_V
1817rabex 4267 . . . 4  |-  { x  e.  B  |  E. p  e.  A  p C x }  e.  _V
1914, 15, 18fvmpt 5709 . . 3  |-  ( K  e.  _V  ->  ( LLines `
 K )  =  { x  e.  B  |  E. p  e.  A  p C x } )
202, 19syl5eq 2410 . 2  |-  ( K  e.  _V  ->  N  =  { x  e.  B  |  E. p  e.  A  p C x } )
211, 20syl 15 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 1647    e. wcel 1715   E.wrex 2629   {crab 2632   _Vcvv 2873   class class class wbr 4125   ` cfv 5358   Basecbs 13356    <o ccvr 29523   Atomscatm 29524   LLinesclln 29751
This theorem is referenced by:  islln  29766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-llines 29758
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