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Theorem lineset 30549
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 2809 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 lineset.n . . 3  |-  N  =  ( Lines `  K )
3 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 lineset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2346 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5541 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
7 lineset.l . . . . . . . . . . . . 13  |-  .<_  =  ( le `  K )
86, 7syl6eqr 2346 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
98breqd 4050 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
p ( le `  k ) ( q ( join `  k
) r )  <->  p  .<_  ( q ( join `  k
) r ) ) )
10 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
11 lineset.j . . . . . . . . . . . . . 14  |-  .\/  =  ( join `  K )
1210, 11syl6eqr 2346 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
1312oveqd 5891 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
q ( join `  k
) r )  =  ( q  .\/  r
) )
1413breq2d 4051 . . . . . . . . . . 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 2796 . . . . . . . . 9  |-  ( k  =  K  ->  { p  e.  ( Atoms `  k )  |  p ( le `  k ) ( q ( join `  k
) r ) }  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) } )
1716eqeq2d 2307 . . . . . . . 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 2762 . . . . . 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 2762 . . . . 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 2410 . . . 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 30312 . . . 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 5555 . . . . . 6  |-  ( Atoms `  K )  e.  _V
244, 23eqeltri 2366 . . . . 5  |-  A  e. 
_V
25 df-sn 3659 . . . . . . 7  |-  { {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }  =  { s  |  s  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) } }
26 snex 4232 . . . . . . 7  |-  { {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }  e.  _V
2725, 26eqeltrri 2367 . . . . . 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 3258 . . . . . 6  |-  { s  |  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  C_  { s  |  s  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }
3027, 29ssexi 4175 . . . . 5  |-  { s  |  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  e.  _V
3124, 24, 30ab2rexex2 6017 . . . 4  |-  { s  |  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  e.  _V
3221, 22, 31fvmpt 5618 . . 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 2340 . 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 1632    e. wcel 1696   {cab 2282    =/= wne 2459   E.wrex 2557   {crab 2560   _Vcvv 2801   {csn 3653   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   Atomscatm 30075   Linesclines 30305
This theorem is referenced by:  isline  30550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-lines 30312
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