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Theorem islinei 30551
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 959 . . 3  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  Q  e.  A )
2 simpl3 960 . . 3  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  R  e.  A )
3 simpr 447 . . 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 2467 . . . . 5  |-  ( q  =  Q  ->  (
q  =/=  r  <->  Q  =/=  r ) )
5 oveq1 5881 . . . . . . . 8  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
65breq2d 4051 . . . . . . 7  |-  ( q  =  Q  ->  (
p  .<_  ( q  .\/  r )  <->  p  .<_  ( Q  .\/  r ) ) )
76rabbidv 2793 . . . . . 6  |-  ( q  =  Q  ->  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  =  {
p  e.  A  |  p  .<_  ( Q  .\/  r ) } )
87eqeq2d 2307 . . . . 5  |-  ( q  =  Q  ->  ( X  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  <->  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  r ) } ) )
94, 8anbi12d 691 . . . 4  |-  ( q  =  Q  ->  (
( q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } )  <-> 
( Q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( Q  .\/  r ) } ) ) )
10 neeq2 2468 . . . . 5  |-  ( r  =  R  ->  ( Q  =/=  r  <->  Q  =/=  R ) )
11 oveq2 5882 . . . . . . . 8  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1211breq2d 4051 . . . . . . 7  |-  ( r  =  R  ->  (
p  .<_  ( Q  .\/  r )  <->  p  .<_  ( Q  .\/  R ) ) )
1312rabbidv 2793 . . . . . 6  |-  ( r  =  R  ->  { p  e.  A  |  p  .<_  ( Q  .\/  r
) }  =  {
p  e.  A  |  p  .<_  ( Q  .\/  R ) } )
1413eqeq2d 2307 . . . . 5  |-  ( r  =  R  ->  ( X  =  { p  e.  A  |  p  .<_  ( Q  .\/  r
) }  <->  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )
1510, 14anbi12d 691 . . . 4  |-  ( r  =  R  ->  (
( Q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( Q  .\/  r ) } )  <-> 
( Q  =/=  R  /\  X  =  {
p  e.  A  |  p  .<_  ( Q  .\/  R ) } ) ) )
169, 15rspc2ev 2905 . . 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 1182 . 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 958 . . 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 30550 . . 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 15 . 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 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   Atomscatm 30075   Linesclines 30305
This theorem is referenced by:  linepmap  30586
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-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-lines 30312
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