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Theorem isline2 35599
Description: Definition of line in terms of projective map. (Contributed by NM, 25-Jan-2012.)
Hypotheses
Ref Expression
isline2.j  |-  .\/  =  ( join `  K )
isline2.a  |-  A  =  ( Atoms `  K )
isline2.n  |-  N  =  ( Lines `  K )
isline2.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
isline2  |-  ( K  e.  Lat  ->  ( X  e.  N  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  ( M `  (
p  .\/  q )
) ) ) )
Distinct variable groups:    q, p, A    K, p, q    X, p, q
Allowed substitution hints:    .\/ ( q, p)    M( q, p)    N( q, p)

Proof of Theorem isline2
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 eqid 2457 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 isline2.j . . 3  |-  .\/  =  ( join `  K )
3 isline2.a . . 3  |-  A  =  ( Atoms `  K )
4 isline2.n . . 3  |-  N  =  ( Lines `  K )
51, 2, 3, 4isline 35564 . 2  |-  ( K  e.  Lat  ->  ( X  e.  N  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  { r  e.  A  |  r ( le
`  K ) ( p  .\/  q ) } ) ) )
6 simpl 457 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( p  e.  A  /\  q  e.  A
) )  ->  K  e.  Lat )
7 eqid 2457 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
87, 3atbase 35115 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
98ad2antrl 727 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( p  e.  A  /\  q  e.  A
) )  ->  p  e.  ( Base `  K
) )
107, 3atbase 35115 . . . . . . . 8  |-  ( q  e.  A  ->  q  e.  ( Base `  K
) )
1110ad2antll 728 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( p  e.  A  /\  q  e.  A
) )  ->  q  e.  ( Base `  K
) )
127, 2latjcl 15807 . . . . . . 7  |-  ( ( K  e.  Lat  /\  p  e.  ( Base `  K )  /\  q  e.  ( Base `  K
) )  ->  (
p  .\/  q )  e.  ( Base `  K
) )
136, 9, 11, 12syl3anc 1228 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( p  e.  A  /\  q  e.  A
) )  ->  (
p  .\/  q )  e.  ( Base `  K
) )
14 isline2.m . . . . . . 7  |-  M  =  ( pmap `  K
)
157, 1, 3, 14pmapval 35582 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( p  .\/  q )  e.  ( Base `  K
) )  ->  ( M `  ( p  .\/  q ) )  =  { r  e.  A  |  r ( le
`  K ) ( p  .\/  q ) } )
1613, 15syldan 470 . . . . 5  |-  ( ( K  e.  Lat  /\  ( p  e.  A  /\  q  e.  A
) )  ->  ( M `  ( p  .\/  q ) )  =  { r  e.  A  |  r ( le
`  K ) ( p  .\/  q ) } )
1716eqeq2d 2471 . . . 4  |-  ( ( K  e.  Lat  /\  ( p  e.  A  /\  q  e.  A
) )  ->  ( X  =  ( M `  ( p  .\/  q
) )  <->  X  =  { r  e.  A  |  r ( le
`  K ) ( p  .\/  q ) } ) )
1817anbi2d 703 . . 3  |-  ( ( K  e.  Lat  /\  ( p  e.  A  /\  q  e.  A
) )  ->  (
( p  =/=  q  /\  X  =  ( M `  ( p  .\/  q ) ) )  <-> 
( p  =/=  q  /\  X  =  {
r  e.  A  | 
r ( le `  K ) ( p 
.\/  q ) } ) ) )
19182rexbidva 2974 . 2  |-  ( K  e.  Lat  ->  ( E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  ( M `  ( p  .\/  q ) ) )  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  {
r  e.  A  | 
r ( le `  K ) ( p 
.\/  q ) } ) ) )
205, 19bitr4d 256 1  |-  ( K  e.  Lat  ->  ( X  e.  N  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  ( M `  (
p  .\/  q )
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   {crab 2811   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14643   lecple 14718   joincjn 15699   Latclat 15801   Atomscatm 35089   Linesclines 35319   pmapcpmap 35322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-lat 15802  df-ats 35093  df-lines 35326  df-pmap 35329
This theorem is referenced by:  isline3  35601  lncvrelatN  35606  linepsubclN  35776
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