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Theorem lineset 33041
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 3087 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 lineset.n . . 3  |-  N  =  ( Lines `  K )
3 fveq2 5872 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 lineset.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2479 . . . . . 6  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
6 fveq2 5872 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
7 lineset.l . . . . . . . . . . . . 13  |-  .<_  =  ( le `  K )
86, 7syl6eqr 2479 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
98breqd 4428 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
p ( le `  k ) ( q ( join `  k
) r )  <->  p  .<_  ( q ( join `  k
) r ) ) )
10 fveq2 5872 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
11 lineset.j . . . . . . . . . . . . . 14  |-  .\/  =  ( join `  K )
1210, 11syl6eqr 2479 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
1312oveqd 6313 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
q ( join `  k
) r )  =  ( q  .\/  r
) )
1413breq2d 4429 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
p  .<_  ( q (
join `  k )
r )  <->  p  .<_  ( q  .\/  r ) ) )
159, 14bitrd 256 . . . . . . . . . 10  |-  ( k  =  K  ->  (
p ( le `  k ) ( q ( join `  k
) r )  <->  p  .<_  ( q  .\/  r ) ) )
165, 15rabeqbidv 3073 . . . . . . . . 9  |-  ( k  =  K  ->  { p  e.  ( Atoms `  k )  |  p ( le `  k ) ( q ( join `  k
) r ) }  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) } )
1716eqeq2d 2434 . . . . . . . 8  |-  ( k  =  K  ->  (
s  =  { p  e.  ( Atoms `  k )  |  p ( le `  k ) ( q ( join `  k
) r ) }  <-> 
s  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } ) )
1817anbi2d 708 . . . . . . 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 3038 . . . . . 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 3038 . . . . 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 2556 . . . 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 32804 . . . 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 5882 . . . . . 6  |-  ( Atoms `  K )  e.  _V
244, 23eqeltri 2504 . . . . 5  |-  A  e. 
_V
25 df-sn 3994 . . . . . . 7  |-  { {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }  =  { s  |  s  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) } }
26 snex 4654 . . . . . . 7  |-  { {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }  e.  _V
2725, 26eqeltrri 2505 . . . . . 6  |-  { s  |  s  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }  e.  _V
28 simpr 462 . . . . . . 7  |-  ( ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } )  -> 
s  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } )
2928ss2abi 3530 . . . . . 6  |-  { s  |  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  C_  { s  |  s  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } }
3027, 29ssexi 4561 . . . . 5  |-  { s  |  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  e.  _V
3124, 24, 30ab2rexex2 6790 . . . 4  |-  { s  |  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  s  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) }  e.  _V
3221, 22, 31fvmpt 5955 . . 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 2473 . 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 17 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   {cab 2405    =/= wne 2616   E.wrex 2774   {crab 2777   _Vcvv 3078   {csn 3993   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   lecple 15149   joincjn 16133   Atomscatm 32567   Linesclines 32797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-lines 32804
This theorem is referenced by:  isline  33042
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