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Theorem islinei 33025
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 1009 . . 3  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  Q  e.  A )
2 simpl3 1010 . . 3  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  R  e.  A )
3 simpr 462 . . 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 2712 . . . . 5  |-  ( q  =  Q  ->  (
q  =/=  r  <->  Q  =/=  r ) )
5 oveq1 6312 . . . . . . . 8  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
65breq2d 4438 . . . . . . 7  |-  ( q  =  Q  ->  (
p  .<_  ( q  .\/  r )  <->  p  .<_  ( Q  .\/  r ) ) )
76rabbidv 3079 . . . . . 6  |-  ( q  =  Q  ->  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  =  {
p  e.  A  |  p  .<_  ( Q  .\/  r ) } )
87eqeq2d 2443 . . . . 5  |-  ( q  =  Q  ->  ( X  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  <->  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  r ) } ) )
94, 8anbi12d 715 . . . 4  |-  ( q  =  Q  ->  (
( q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } )  <-> 
( Q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( Q  .\/  r ) } ) ) )
10 neeq2 2714 . . . . 5  |-  ( r  =  R  ->  ( Q  =/=  r  <->  Q  =/=  R ) )
11 oveq2 6313 . . . . . . . 8  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1211breq2d 4438 . . . . . . 7  |-  ( r  =  R  ->  (
p  .<_  ( Q  .\/  r )  <->  p  .<_  ( Q  .\/  R ) ) )
1312rabbidv 3079 . . . . . 6  |-  ( r  =  R  ->  { p  e.  A  |  p  .<_  ( Q  .\/  r
) }  =  {
p  e.  A  |  p  .<_  ( Q  .\/  R ) } )
1413eqeq2d 2443 . . . . 5  |-  ( r  =  R  ->  ( X  =  { p  e.  A  |  p  .<_  ( Q  .\/  r
) }  <->  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )
1510, 14anbi12d 715 . . . 4  |-  ( r  =  R  ->  (
( Q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( Q  .\/  r ) } )  <-> 
( Q  =/=  R  /\  X  =  {
p  e.  A  |  p  .<_  ( Q  .\/  R ) } ) ) )
169, 15rspc2ev 3199 . . 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 1264 . 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 1008 . . 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 33024 . . 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 17 . 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 235 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783   {crab 2786   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   lecple 15160   joincjn 16144   Atomscatm 32549   Linesclines 32779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-lines 32786
This theorem is referenced by:  linepmap  33060
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