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Theorem linepmap 30634
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 961 . . 3  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  Lat )
2 simpl2 962 . . . . 5  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  A )
3 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
4 isline2.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4atbase 30149 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
62, 5syl 16 . . . 4  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  ( Base `  K )
)
7 simpl3 963 . . . . 5  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
83, 4atbase 30149 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
97, 8syl 16 . . . 4  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  ( Base `  K )
)
10 isline2.j . . . . 5  |-  .\/  =  ( join `  K )
113, 10latjcl 14481 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
121, 6, 9, 11syl3anc 1185 . . 3  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
13 eqid 2438 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
14 isline2.m . . . 4  |-  M  =  ( pmap `  K
)
153, 13, 4, 14pmapval 30616 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( M `  ( P  .\/  Q ) )  =  { r  e.  A  |  r ( le
`  K ) ( P  .\/  Q ) } )
161, 12, 15syl2anc 644 . 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 2438 . . 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 30599 . . 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 667 . 2  |-  ( ( ( K  e.  Lat  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  { r  e.  A  |  r
( le `  K
) ( P  .\/  Q ) }  e.  N
)
2116, 20eqeltrd 2512 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   Latclat 14476   Atomscatm 30123   Linesclines 30353   pmapcpmap 30356
This theorem is referenced by:  cdleme3h  31094  cdleme7ga  31107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-lat 14477  df-ats 30127  df-lines 30360  df-pmap 30363
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