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Theorem linepmap 30586
Description: A line described with a projective map. (Contributed by NM, 3-Feb-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
linepmap  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( M `  ( P  .\/  Q
) )  e.  N
)

Proof of Theorem linepmap
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 simpl1 958 . . 3  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  Lat )
2 simpl2 959 . . . . 5  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  A )
3 eqid 2296 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
4 isline2.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4atbase 30101 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
62, 5syl 15 . . . 4  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  ( Base `  K )
)
7 simpl3 960 . . . . 5  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
83, 4atbase 30101 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
97, 8syl 15 . . . 4  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  ( Base `  K )
)
10 isline2.j . . . . 5  |-  .\/  =  ( join `  K )
113, 10latjcl 14172 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
121, 6, 9, 11syl3anc 1182 . . 3  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
13 eqid 2296 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
14 isline2.m . . . 4  |-  M  =  ( pmap `  K
)
153, 13, 4, 14pmapval 30568 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( M `  ( P  .\/  Q ) )  =  { r  e.  A  |  r ( le
`  K ) ( P  .\/  Q ) } )
161, 12, 15syl2anc 642 . 2  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( M `  ( P  .\/  Q
) )  =  {
r  e.  A  | 
r ( le `  K ) ( P 
.\/  Q ) } )
17 eqid 2296 . . 3  |-  { r  e.  A  |  r ( le `  K
) ( P  .\/  Q ) }  =  {
r  e.  A  | 
r ( le `  K ) ( P 
.\/  Q ) }
18 isline2.n . . . 4  |-  N  =  ( Lines `  K )
1913, 10, 4, 18islinei 30551 . . 3  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  ( P  =/=  Q  /\  { r  e.  A  |  r ( le
`  K ) ( P  .\/  Q ) }  =  { r  e.  A  |  r ( le `  K
) ( P  .\/  Q ) } ) )  ->  { r  e.  A  |  r ( le `  K ) ( P  .\/  Q
) }  e.  N
)
2017, 19mpanr2 665 . 2  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  { r  e.  A  |  r
( le `  K
) ( P  .\/  Q ) }  e.  N
)
2116, 20eqeltrd 2370 1  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( M `  ( P  .\/  Q
) )  e.  N
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Latclat 14167   Atomscatm 30075   Linesclines 30305   pmapcpmap 30308
This theorem is referenced by:  cdleme3h  31046  cdleme7ga  31059
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-pow 4204  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-lat 14168  df-ats 30079  df-lines 30312  df-pmap 30315
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