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Theorem lineset 29300
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 2796 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 lineset.n . . 3  |-  N  =  ( Lines `  K )
3 fveq2 5525 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 lineset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2333 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5525 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
7 lineset.l . . . . . . . . . . . . 13  |-  .<_  =  ( le `  K )
86, 7syl6eqr 2333 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
98breqd 4034 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
p ( le `  k ) ( q ( join `  k
) r )  <->  p  .<_  ( q ( join `  k
) r ) ) )
10 fveq2 5525 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
11 lineset.j . . . . . . . . . . . . . 14  |-  .\/  =  ( join `  K )
1210, 11syl6eqr 2333 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
1312oveqd 5875 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
q ( join `  k
) r )  =  ( q  .\/  r
) )
1413breq2d 4035 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
p  .<_  ( q (
join `  k )
r )  <->  p  .<_  ( q  .\/  r ) ) )
159, 14bitrd 244 . . . . . . . . . 10  |-  ( k  =  K  ->  (
p ( le `  k ) ( q ( join `  k
) r )  <->  p  .<_  ( q  .\/  r ) ) )
165, 15rabeqbidv 2783 . . . . . . . . 9  |-  ( k  =  K  ->  { p  e.  ( Atoms `  k )  |  p ( le `  k ) ( q ( join `  k
) r ) }  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) } )
1716eqeq2d 2294 . . . . . . . 8  |-  ( k  =  K  ->  (
s  =  { p  e.  ( Atoms `  k )  |  p ( le `  k ) ( q ( join `  k
) r ) }  <-> 
s  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } ) )
1817anbi2d 684 . . . . . . 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 2749 . . . . . 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 2749 . . . . 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 2397 . . . 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 29063 . . . 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 5539 . . . . . 6  |-  ( Atoms `  K )  e.  _V
244, 23eqeltri 2353 . . . . 5  |-  A  e. 
_V
25 df-sn 3646 . . . . . . 7  |-  { {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }  =  { s  |  s  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) } }
26 snex 4216 . . . . . . 7  |-  { {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }  e.  _V
2725, 26eqeltrri 2354 . . . . . 6  |-  { s  |  s  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }  e.  _V
28 simpr 447 . . . . . . 7  |-  ( ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } )  -> 
s  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } )
2928ss2abi 3245 . . . . . 6  |-  { s  |  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  C_  { s  |  s  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }
3027, 29ssexi 4159 . . . . 5  |-  { s  |  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  e.  _V
3124, 24, 30ab2rexex2 6001 . . . 4  |-  { s  |  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  e.  _V
3221, 22, 31fvmpt 5602 . . 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 2327 . 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 15 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 358    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   E.wrex 2544   {crab 2547   _Vcvv 2788   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 28826   Linesclines 29056
This theorem is referenced by:  isline  29301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-lines 29063
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