Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  linepmap Unicode version

Theorem linepmap 29964
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 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
4 isline2.a . . . . . 6  |-  A  =  ( Atoms `  K )
53, 4atbase 29479 . . . . 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 29479 . . . . 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 14156 . . . 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 2283 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
14 isline2.m . . . 4  |-  M  =  ( pmap `  K
)
153, 13, 4, 14pmapval 29946 . . 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 2283 . . 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 29929 . . 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 2357 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 1623    e. wcel 1684    =/= wne 2446   {crab 2547   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 29453   Linesclines 29683   pmapcpmap 29686
This theorem is referenced by:  cdleme3h  30424  cdleme7ga  30437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-lat 14152  df-ats 29457  df-lines 29690  df-pmap 29693
  Copyright terms: Public domain W3C validator