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Theorem lineset 29853
Description: The set of lines in a Hilbert lattice. (Contributed by NM, 19-Sep-2011.)
Hypotheses
Ref Expression
lineset.l  |-  .<_  =  ( le `  K )
lineset.j  |-  .\/  =  ( join `  K )
lineset.a  |-  A  =  ( Atoms `  K )
lineset.n  |-  N  =  ( Lines `  K )
Assertion
Ref Expression
lineset  |-  ( K  e.  B  ->  N  =  { s  |  E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } ) } )
Distinct variable groups:    q, p, r, s, A    K, p, q, r, s    .\/ , s    .<_ , s
Allowed substitution hints:    B( s, r, q, p)    .\/ ( r, q, p)    .<_ ( r, q, p)    N( s, r, q, p)

Proof of Theorem lineset
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2908 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 lineset.n . . 3  |-  N  =  ( Lines `  K )
3 fveq2 5669 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 lineset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2438 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5669 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
7 lineset.l . . . . . . . . . . . . 13  |-  .<_  =  ( le `  K )
86, 7syl6eqr 2438 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
98breqd 4165 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
p ( le `  k ) ( q ( join `  k
) r )  <->  p  .<_  ( q ( join `  k
) r ) ) )
10 fveq2 5669 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
11 lineset.j . . . . . . . . . . . . . 14  |-  .\/  =  ( join `  K )
1210, 11syl6eqr 2438 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
1312oveqd 6038 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
q ( join `  k
) r )  =  ( q  .\/  r
) )
1413breq2d 4166 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
p  .<_  ( q (
join `  k )
r )  <->  p  .<_  ( q  .\/  r ) ) )
159, 14bitrd 245 . . . . . . . . . 10  |-  ( k  =  K  ->  (
p ( le `  k ) ( q ( join `  k
) r )  <->  p  .<_  ( q  .\/  r ) ) )
165, 15rabeqbidv 2895 . . . . . . . . 9  |-  ( k  =  K  ->  { p  e.  ( Atoms `  k )  |  p ( le `  k ) ( q ( join `  k
) r ) }  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) } )
1716eqeq2d 2399 . . . . . . . 8  |-  ( k  =  K  ->  (
s  =  { p  e.  ( Atoms `  k )  |  p ( le `  k ) ( q ( join `  k
) r ) }  <-> 
s  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } ) )
1817anbi2d 685 . . . . . . 7  |-  ( k  =  K  ->  (
( q  =/=  r  /\  s  =  {
p  e.  ( Atoms `  k )  |  p ( le `  k
) ( q (
join `  k )
r ) } )  <-> 
( q  =/=  r  /\  s  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } ) ) )
195, 18rexeqbidv 2861 . . . . . 6  |-  ( k  =  K  ->  ( E. r  e.  ( Atoms `  k ) ( q  =/=  r  /\  s  =  { p  e.  ( Atoms `  k )  |  p ( le `  k ) ( q ( join `  k
) r ) } )  <->  E. r  e.  A  ( q  =/=  r  /\  s  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } ) ) )
205, 19rexeqbidv 2861 . . . . 5  |-  ( k  =  K  ->  ( E. q  e.  ( Atoms `  k ) E. r  e.  ( Atoms `  k ) ( q  =/=  r  /\  s  =  { p  e.  (
Atoms `  k )  |  p ( le `  k ) ( q ( join `  k
) r ) } )  <->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  s  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } ) ) )
2120abbidv 2502 . . . 4  |-  ( k  =  K  ->  { s  |  E. q  e.  ( Atoms `  k ) E. r  e.  ( Atoms `  k ) ( q  =/=  r  /\  s  =  { p  e.  ( Atoms `  k )  |  p ( le `  k ) ( q ( join `  k
) r ) } ) }  =  {
s  |  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) } )
22 df-lines 29616 . . . 4  |-  Lines  =  ( k  e.  _V  |->  { s  |  E. q  e.  ( Atoms `  k ) E. r  e.  ( Atoms `  k ) ( q  =/=  r  /\  s  =  { p  e.  ( Atoms `  k )  |  p ( le `  k ) ( q ( join `  k
) r ) } ) } )
23 fvex 5683 . . . . . 6  |-  ( Atoms `  K )  e.  _V
244, 23eqeltri 2458 . . . . 5  |-  A  e. 
_V
25 df-sn 3764 . . . . . . 7  |-  { {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }  =  { s  |  s  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) } }
26 snex 4347 . . . . . . 7  |-  { {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }  e.  _V
2725, 26eqeltrri 2459 . . . . . 6  |-  { s  |  s  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }  e.  _V
28 simpr 448 . . . . . . 7  |-  ( ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } )  -> 
s  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } )
2928ss2abi 3359 . . . . . 6  |-  { s  |  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  C_  { s  |  s  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }
3027, 29ssexi 4290 . . . . 5  |-  { s  |  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  e.  _V
3124, 24, 30ab2rexex2 6167 . . . 4  |-  { s  |  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  e.  _V
3221, 22, 31fvmpt 5746 . . 3  |-  ( K  e.  _V  ->  ( Lines `  K )  =  { s  |  E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } ) } )
332, 32syl5eq 2432 . 2  |-  ( K  e.  _V  ->  N  =  { s  |  E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } ) } )
341, 33syl 16 1  |-  ( K  e.  B  ->  N  =  { s  |  E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2374    =/= wne 2551   E.wrex 2651   {crab 2654   _Vcvv 2900   {csn 3758   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   lecple 13464   joincjn 14329   Atomscatm 29379   Linesclines 29609
This theorem is referenced by:  isline  29854
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-lines 29616
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