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Theorem islinei 33480
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 992 . . 3  |-  ( ( ( K  e.  D  /\  Q  e.  A  /\  R  e.  A
)  /\  ( Q  =/=  R  /\  X  =  { p  e.  A  |  p  .<_  ( Q 
.\/  R ) } ) )  ->  Q  e.  A )
2 simpl3 993 . . 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 2644 . . . . 5  |-  ( q  =  Q  ->  (
q  =/=  r  <->  Q  =/=  r ) )
5 oveq1 6119 . . . . . . . 8  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
65breq2d 4325 . . . . . . 7  |-  ( q  =  Q  ->  (
p  .<_  ( q  .\/  r )  <->  p  .<_  ( Q  .\/  r ) ) )
76rabbidv 2985 . . . . . 6  |-  ( q  =  Q  ->  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  =  {
p  e.  A  |  p  .<_  ( Q  .\/  r ) } )
87eqeq2d 2454 . . . . 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 2646 . . . . 5  |-  ( r  =  R  ->  ( Q  =/=  r  <->  Q  =/=  R ) )
11 oveq2 6120 . . . . . . . 8  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1211breq2d 4325 . . . . . . 7  |-  ( r  =  R  ->  (
p  .<_  ( Q  .\/  r )  <->  p  .<_  ( Q  .\/  R ) ) )
1312rabbidv 2985 . . . . . 6  |-  ( r  =  R  ->  { p  e.  A  |  p  .<_  ( Q  .\/  r
) }  =  {
p  e.  A  |  p  .<_  ( Q  .\/  R ) } )
1413eqeq2d 2454 . . . . 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 3102 . . 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 1218 . 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 991 . . 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 33479 . . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620   E.wrex 2737   {crab 2740   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   lecple 14266   joincjn 15135   Atomscatm 33004   Linesclines 33234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-lines 33241
This theorem is referenced by:  linepmap  33515
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