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Theorem isline2 33415
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 2441 . . 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 33380 . 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 2441 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
87, 3atbase 32931 . . . . . . . 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 32931 . . . . . . . 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 15219 . . . . . . 7  |-  ( ( K  e.  Lat  /\  p  e.  ( Base `  K )  /\  q  e.  ( Base `  K
) )  ->  (
p  .\/  q )  e.  ( Base `  K
) )
136, 9, 11, 12syl3anc 1218 . . . . . 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 33398 . . . . . 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 2452 . . . 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 2754 . 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 1369    e. wcel 1756    =/= wne 2604   E.wrex 2714   {crab 2717   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   Basecbs 14172   lecple 14243   joincjn 15112   Latclat 15213   Atomscatm 32905   Linesclines 33135   pmapcpmap 33138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-lub 15142  df-glb 15143  df-join 15144  df-meet 15145  df-lat 15214  df-ats 32909  df-lines 33142  df-pmap 33145
This theorem is referenced by:  isline3  33417  lncvrelatN  33422  linepsubclN  33592
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