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

Theorem islinei 30538
Description: Condition implying "is a line". (Contributed by NM, 3-Feb-2012.)
Hypotheses
Ref Expression
isline.l  |-  .<_  =  ( le `  K )
isline.j  |-  .\/  =  ( join `  K )
isline.a  |-  A  =  ( Atoms `  K )
isline.n  |-  N  =  ( Lines `  K )
Assertion
Ref Expression
islinei  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  X  e.  N )
Distinct variable groups:    A, p    K, p    Q, p    R, p
Allowed substitution hints:    D( p)    .\/ ( p)    .<_ ( p)    N( p)    X( p)

Proof of Theorem islinei
Dummy variables  q 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 962 . . 3  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  Q  e.  A )
2 simpl3 963 . . 3  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  R  e.  A )
3 simpr 449 . . 3  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )
4 neeq1 2610 . . . . 5  |-  ( q  =  Q  ->  (
q  =/=  r  <->  Q  =/=  r ) )
5 oveq1 6089 . . . . . . . 8  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
65breq2d 4225 . . . . . . 7  |-  ( q  =  Q  ->  (
p  .<_  ( q  .\/  r )  <->  p  .<_  ( Q  .\/  r ) ) )
76rabbidv 2949 . . . . . 6  |-  ( q  =  Q  ->  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  =  {
p  e.  A  |  p  .<_  ( Q  .\/  r ) } )
87eqeq2d 2448 . . . . 5  |-  ( q  =  Q  ->  ( X  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  <->  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  r ) } ) )
94, 8anbi12d 693 . . . 4  |-  ( q  =  Q  ->  (
( q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } )  <-> 
( Q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( Q  .\/  r ) } ) ) )
10 neeq2 2611 . . . . 5  |-  ( r  =  R  ->  ( Q  =/=  r  <->  Q  =/=  R ) )
11 oveq2 6090 . . . . . . . 8  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1211breq2d 4225 . . . . . . 7  |-  ( r  =  R  ->  (
p  .<_  ( Q  .\/  r )  <->  p  .<_  ( Q  .\/  R ) ) )
1312rabbidv 2949 . . . . . 6  |-  ( r  =  R  ->  { p  e.  A  |  p  .<_  ( Q  .\/  r
) }  =  {
p  e.  A  |  p  .<_  ( Q  .\/  R ) } )
1413eqeq2d 2448 . . . . 5  |-  ( r  =  R  ->  ( X  =  { p  e.  A  |  p  .<_  ( Q  .\/  r
) }  <->  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )
1510, 14anbi12d 693 . . . 4  |-  ( r  =  R  ->  (
( Q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( Q  .\/  r ) } )  <-> 
( Q  =/=  R  /\  X  =  {
p  e.  A  |  p  .<_  ( Q  .\/  R ) } ) ) )
169, 15rspc2ev 3061 . . 3  |-  ( ( Q  e.  A  /\  R  e.  A  /\  ( Q  =/=  R  /\  X  =  {
p  e.  A  |  p  .<_  ( Q  .\/  R ) } ) )  ->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } ) )
171, 2, 3, 16syl3anc 1185 . 2  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) )
18 simpl1 961 . . 3  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  K  e.  D )
19 isline.l . . . 4  |-  .<_  =  ( le `  K )
20 isline.j . . . 4  |-  .\/  =  ( join `  K )
21 isline.a . . . 4  |-  A  =  ( Atoms `  K )
22 isline.n . . . 4  |-  N  =  ( Lines `  K )
2319, 20, 21, 22isline 30537 . . 3  |-  ( K  e.  D  ->  ( X  e.  N  <->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) ) )
2418, 23syl 16 . 2  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  ( X  e.  N  <->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) ) )
2517, 24mpbird 225 1  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  X  e.  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707   {crab 2710   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   lecple 13537   joincjn 14402   Atomscatm 30062   Linesclines 30292
This theorem is referenced by:  linepmap  30573
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-lines 30299
  Copyright terms: Public domain W3C validator