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Theorem lineset 30472
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 2956 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 lineset.n . . 3  |-  N  =  ( Lines `  K )
3 fveq2 5720 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 lineset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2485 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5720 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
7 lineset.l . . . . . . . . . . . . 13  |-  .<_  =  ( le `  K )
86, 7syl6eqr 2485 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
98breqd 4215 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
p ( le `  k ) ( q ( join `  k
) r )  <->  p  .<_  ( q ( join `  k
) r ) ) )
10 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
11 lineset.j . . . . . . . . . . . . . 14  |-  .\/  =  ( join `  K )
1210, 11syl6eqr 2485 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
1312oveqd 6090 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
q ( join `  k
) r )  =  ( q  .\/  r
) )
1413breq2d 4216 . . . . . . . . . . 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 2943 . . . . . . . . 9  |-  ( k  =  K  ->  { p  e.  ( Atoms `  k )  |  p ( le `  k ) ( q ( join `  k
) r ) }  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) } )
1716eqeq2d 2446 . . . . . . . 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 2909 . . . . . 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 2909 . . . . 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 2549 . . . 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 30235 . . . 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 5734 . . . . . 6  |-  ( Atoms `  K )  e.  _V
244, 23eqeltri 2505 . . . . 5  |-  A  e. 
_V
25 df-sn 3812 . . . . . . 7  |-  { {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }  =  { s  |  s  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) } }
26 snex 4397 . . . . . . 7  |-  { {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }  e.  _V
2725, 26eqeltrri 2506 . . . . . 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 3407 . . . . . 6  |-  { s  |  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  C_  { s  |  s  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }
3027, 29ssexi 4340 . . . . 5  |-  { s  |  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  e.  _V
3124, 24, 30ab2rexex2 6219 . . . 4  |-  { s  |  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  e.  _V
3221, 22, 31fvmpt 5798 . . 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 2479 . 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 1652    e. wcel 1725   {cab 2421    =/= wne 2598   E.wrex 2698   {crab 2701   _Vcvv 2948   {csn 3806   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   Atomscatm 29998   Linesclines 30228
This theorem is referenced by:  isline  30473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-lines 30235
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