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Theorem islinei 34937
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 1000 . . 3  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  Q  e.  A )
2 simpl3 1001 . . 3  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  R  e.  A )
3 simpr 461 . . 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 2748 . . . . 5  |-  ( q  =  Q  ->  (
q  =/=  r  <->  Q  =/=  r ) )
5 oveq1 6302 . . . . . . . 8  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
65breq2d 4465 . . . . . . 7  |-  ( q  =  Q  ->  (
p  .<_  ( q  .\/  r )  <->  p  .<_  ( Q  .\/  r ) ) )
76rabbidv 3110 . . . . . 6  |-  ( q  =  Q  ->  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  =  {
p  e.  A  |  p  .<_  ( Q  .\/  r ) } )
87eqeq2d 2481 . . . . 5  |-  ( q  =  Q  ->  ( X  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  <->  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  r ) } ) )
94, 8anbi12d 710 . . . 4  |-  ( q  =  Q  ->  (
( q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } )  <-> 
( Q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( Q  .\/  r ) } ) ) )
10 neeq2 2750 . . . . 5  |-  ( r  =  R  ->  ( Q  =/=  r  <->  Q  =/=  R ) )
11 oveq2 6303 . . . . . . . 8  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1211breq2d 4465 . . . . . . 7  |-  ( r  =  R  ->  (
p  .<_  ( Q  .\/  r )  <->  p  .<_  ( Q  .\/  R ) ) )
1312rabbidv 3110 . . . . . 6  |-  ( r  =  R  ->  { p  e.  A  |  p  .<_  ( Q  .\/  r
) }  =  {
p  e.  A  |  p  .<_  ( Q  .\/  R ) } )
1413eqeq2d 2481 . . . . 5  |-  ( r  =  R  ->  ( X  =  { p  e.  A  |  p  .<_  ( Q  .\/  r
) }  <->  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )
1510, 14anbi12d 710 . . . 4  |-  ( r  =  R  ->  (
( Q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( Q  .\/  r ) } )  <-> 
( Q  =/=  R  /\  X  =  {
p  e.  A  |  p  .<_  ( Q  .\/  R ) } ) ) )
169, 15rspc2ev 3230 . . 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 1228 . 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 999 . . 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 34936 . . 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 232 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   {crab 2821   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   lecple 14579   joincjn 15448   Atomscatm 34461   Linesclines 34691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-lines 34698
This theorem is referenced by:  linepmap  34972
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