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Theorem lineunray 30863
Description: A line is composed of a point and the two rays emerging from it. Theorem 6.15 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
lineunray  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  -> 
( P  Btwn  <. Q ,  R >.  ->  ( PLine Q )  =  ( ( ( PRay Q
)  u.  { P } )  u.  ( PRay R ) ) ) )

Proof of Theorem lineunray
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl1 1008 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  ->  N  e.  NN )
2 simpr 462 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  ->  x  e.  ( EE `  N ) )
3 simpl21 1083 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  ->  P  e.  ( EE `  N ) )
4 simpl22 1084 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  ->  Q  e.  ( EE `  N ) )
5 brcolinear 30775 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  ( x  e.  ( EE `  N )  /\  P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  -> 
( x  Colinear  <. P ,  Q >. 
<->  ( x  Btwn  <. P ,  Q >.  \/  P  Btwn  <. Q ,  x >.  \/  Q  Btwn  <. x ,  P >. ) ) )
61, 2, 3, 4, 5syl13anc 1266 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( x  Colinear  <. P ,  Q >. 
<->  ( x  Btwn  <. P ,  Q >.  \/  P  Btwn  <. Q ,  x >.  \/  Q  Btwn  <. x ,  P >. ) ) )
76adantr 466 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( x  Colinear  <. P ,  Q >.  <->  (
x  Btwn  <. P ,  Q >.  \/  P  Btwn  <. Q ,  x >.  \/  Q  Btwn  <. x ,  P >. ) ) )
8 olc 385 . . . . . . . . . . . . . 14  |-  ( x 
Btwn  <. P ,  Q >.  ->  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )
98orcd 393 . . . . . . . . . . . . 13  |-  ( x 
Btwn  <. P ,  Q >.  ->  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
109a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( x  Btwn  <. P ,  Q >.  ->  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
11 simpl3l 1060 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  ->  P  =/=  Q )
1211necomd 2656 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  ->  Q  =/=  P )
1312adantr 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  P  Btwn  <. Q ,  x >. ) )  ->  Q  =/=  P )
14 simprl 762 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  P  Btwn  <. Q ,  x >. ) )  ->  P  Btwn  <. Q ,  R >. )
15 simprr 764 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  P  Btwn  <. Q ,  x >. ) )  ->  P  Btwn  <. Q ,  x >. )
1613, 14, 153jca 1185 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  P  Btwn  <. Q ,  x >. ) )  ->  ( Q  =/=  P  /\  P  Btwn  <. Q ,  R >.  /\  P  Btwn  <. Q ,  x >. ) )
17 simpl23 1085 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  ->  R  e.  ( EE `  N ) )
18 btwnconn2 30818 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) )  /\  ( R  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  ->  ( ( Q  =/=  P  /\  P  Btwn  <. Q ,  R >.  /\  P  Btwn  <. Q ,  x >. )  ->  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
191, 4, 3, 17, 2, 18syl122anc 1273 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( Q  =/= 
P  /\  P  Btwn  <. Q ,  R >.  /\  P  Btwn  <. Q ,  x >. )  ->  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
2019adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  P  Btwn  <. Q ,  x >. ) )  ->  ( ( Q  =/=  P  /\  P  Btwn  <. Q ,  R >.  /\  P  Btwn  <. Q ,  x >. )  ->  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
2116, 20mpd 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  P  Btwn  <. Q ,  x >. ) )  ->  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) )
2221olcd 394 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  P  Btwn  <. Q ,  x >. ) )  ->  ( ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
2322expr 618 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( P  Btwn  <. Q ,  x >.  ->  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
24 btwncom 30730 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  x  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  -> 
( Q  Btwn  <. x ,  P >.  <->  Q  Btwn  <. P ,  x >. ) )
251, 4, 2, 3, 24syl13anc 1266 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( Q  Btwn  <. x ,  P >.  <->  Q  Btwn  <. P ,  x >. ) )
26 orc 386 . . . . . . . . . . . . . . 15  |-  ( Q 
Btwn  <. P ,  x >.  ->  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )
2726orcd 393 . . . . . . . . . . . . . 14  |-  ( Q 
Btwn  <. P ,  x >.  ->  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
2825, 27syl6bi 231 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( Q  Btwn  <. x ,  P >.  ->  ( ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
2928adantr 466 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( Q  Btwn  <. x ,  P >.  ->  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
3010, 23, 293jaod 1328 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( (
x  Btwn  <. P ,  Q >.  \/  P  Btwn  <. Q ,  x >.  \/  Q  Btwn  <. x ,  P >. )  ->  (
( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
317, 30sylbid 218 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( x  Colinear  <. P ,  Q >.  -> 
( ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
32 olc 385 . . . . . . . . . 10  |-  ( ( ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) )  -> 
( x  =  P  \/  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
3331, 32syl6 34 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( x  Colinear  <. P ,  Q >.  -> 
( x  =  P  \/  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) ) )
34 colineartriv1 30783 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) )  ->  P  Colinear  <. P ,  Q >. )
351, 3, 4, 34syl3anc 1264 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  ->  P  Colinear  <. P ,  Q >. )
36 breq1 4369 . . . . . . . . . . . 12  |-  ( x  =  P  ->  (
x  Colinear  <. P ,  Q >.  <-> 
P  Colinear  <. P ,  Q >. ) )
3735, 36syl5ibrcom 225 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( x  =  P  ->  x  Colinear  <. P ,  Q >. ) )
3837adantr 466 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( x  =  P  ->  x  Colinear  <. P ,  Q >. )
)
39 btwncolinear3 30787 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  x  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  -> 
( Q  Btwn  <. P ,  x >.  ->  x  Colinear  <. P ,  Q >. ) )
401, 3, 2, 4, 39syl13anc 1266 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( Q  Btwn  <. P ,  x >.  ->  x  Colinear  <. P ,  Q >. ) )
41 btwncolinear5 30789 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  x  e.  ( EE `  N
) ) )  -> 
( x  Btwn  <. P ,  Q >.  ->  x  Colinear  <. P ,  Q >. ) )
421, 3, 4, 2, 41syl13anc 1266 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( x  Btwn  <. P ,  Q >.  ->  x  Colinear  <. P ,  Q >. ) )
4340, 42jaod 381 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  ->  x  Colinear  <. P ,  Q >. ) )
4443adantr 466 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  ->  x  Colinear  <. P ,  Q >. ) )
45 simpl3r 1061 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  ->  P  =/=  R )
4645adantr 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  R  Btwn  <. P ,  x >. ) )  ->  P  =/=  R )
47 simprl 762 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  R  Btwn  <. P ,  x >. ) )  ->  P  Btwn  <. Q ,  R >. )
48 simprr 764 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  R  Btwn  <. P ,  x >. ) )  ->  R  Btwn  <. P ,  x >. )
4946, 47, 483jca 1185 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  R  Btwn  <. P ,  x >. ) )  ->  ( P  =/=  R  /\  P  Btwn  <. Q ,  R >.  /\  R  Btwn  <. P ,  x >. ) )
50 btwnouttr 30740 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) )  /\  ( R  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  ->  ( ( P  =/=  R  /\  P  Btwn  <. Q ,  R >.  /\  R  Btwn  <. P ,  x >. )  ->  P  Btwn  <. Q ,  x >. ) )
511, 4, 3, 17, 2, 50syl122anc 1273 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( P  =/= 
R  /\  P  Btwn  <. Q ,  R >.  /\  R  Btwn  <. P ,  x >. )  ->  P  Btwn  <. Q ,  x >. ) )
5251adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  R  Btwn  <. P ,  x >. ) )  ->  ( ( P  =/=  R  /\  P  Btwn  <. Q ,  R >.  /\  R  Btwn  <. P ,  x >. )  ->  P  Btwn  <. Q ,  x >. ) )
5349, 52mpd 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  R  Btwn  <. P ,  x >. ) )  ->  P  Btwn  <. Q ,  x >. )
54 btwncolinear4 30788 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  x  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  -> 
( P  Btwn  <. Q ,  x >.  ->  x  Colinear  <. P ,  Q >. ) )
551, 4, 2, 3, 54syl13anc 1266 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( P  Btwn  <. Q ,  x >.  ->  x  Colinear  <. P ,  Q >. ) )
5655adantr 466 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  R  Btwn  <. P ,  x >. ) )  ->  ( P  Btwn  <. Q ,  x >.  ->  x  Colinear  <. P ,  Q >. ) )
5753, 56mpd 15 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  R  Btwn  <. P ,  x >. ) )  ->  x  Colinear  <. P ,  Q >. )
5857expr 618 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( R  Btwn  <. P ,  x >.  ->  x  Colinear  <. P ,  Q >. ) )
59 simprr 764 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  x  Btwn  <. P ,  R >. ) )  ->  x  Btwn  <. P ,  R >. )
601, 2, 3, 17, 59btwncomand 30731 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  x  Btwn  <. P ,  R >. ) )  ->  x  Btwn  <. R ,  P >. )
61 simprl 762 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  x  Btwn  <. P ,  R >. ) )  ->  P  Btwn  <. Q ,  R >. )
621, 3, 4, 17, 61btwncomand 30731 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  x  Btwn  <. P ,  R >. ) )  ->  P  Btwn  <. R ,  Q >. )
631, 17, 2, 3, 4, 60, 62btwnexch3and 30737 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  x  Btwn  <. P ,  R >. ) )  ->  P  Btwn  <.
x ,  Q >. )
64 btwncolinear2 30786 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN  /\  ( x  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  -> 
( P  Btwn  <. x ,  Q >.  ->  x  Colinear  <. P ,  Q >. )
)
651, 2, 4, 3, 64syl13anc 1266 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( P  Btwn  <. x ,  Q >.  ->  x  Colinear  <. P ,  Q >. )
)
6665adantr 466 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  x  Btwn  <. P ,  R >. ) )  ->  ( P  Btwn  <. x ,  Q >.  ->  x  Colinear  <. P ,  Q >. ) )
6763, 66mpd 15 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  ( P  Btwn  <. Q ,  R >.  /\  x  Btwn  <. P ,  R >. ) )  ->  x  Colinear  <. P ,  Q >. )
6867expr 618 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( x  Btwn  <. P ,  R >.  ->  x  Colinear  <. P ,  Q >. ) )
6958, 68jaod 381 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. )  ->  x  Colinear  <. P ,  Q >. ) )
7044, 69jaod 381 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( (
( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) )  ->  x  Colinear  <. P ,  Q >. ) )
7138, 70jaod 381 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( (
x  =  P  \/  ( ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )  ->  x  Colinear  <. P ,  Q >. ) )
7233, 71impbid 193 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( x  Colinear  <. P ,  Q >.  <->  (
x  =  P  \/  ( ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) ) )
73 pm5.63 932 . . . . . . . . 9  |-  ( ( x  =  P  \/  ( ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )  <-> 
( x  =  P  \/  ( -.  x  =  P  /\  (
( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) ) )
74 df-ne 2601 . . . . . . . . . . . 12  |-  ( x  =/=  P  <->  -.  x  =  P )
7574anbi1i 699 . . . . . . . . . . 11  |-  ( ( x  =/=  P  /\  ( ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )  <-> 
( -.  x  =  P  /\  ( ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
76 andi 875 . . . . . . . . . . 11  |-  ( ( x  =/=  P  /\  ( ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )  <-> 
( ( x  =/= 
P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )  \/  ( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
7775, 76bitr3i 254 . . . . . . . . . 10  |-  ( ( -.  x  =  P  /\  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )  <-> 
( ( x  =/= 
P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )  \/  ( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
7877orbi2i 521 . . . . . . . . 9  |-  ( ( x  =  P  \/  ( -.  x  =  P  /\  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )  <->  ( x  =  P  \/  ( ( x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )  \/  ( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) ) )
7973, 78bitri 252 . . . . . . . 8  |-  ( ( x  =  P  \/  ( ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )  <-> 
( x  =  P  \/  ( ( x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )  \/  ( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) ) )
8072, 79syl6bb 264 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( x  Colinear  <. P ,  Q >.  <->  (
x  =  P  \/  ( ( x  =/= 
P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )  \/  ( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) ) ) )
81 broutsideof2 30838 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  x  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. Q ,  x >.  <-> 
( Q  =/=  P  /\  x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) ) ) )
821, 3, 4, 2, 81syl13anc 1266 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( POutsideOf <. Q ,  x >.  <-> 
( Q  =/=  P  /\  x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) ) ) )
83 3simpc 1004 . . . . . . . . . . . 12  |-  ( ( Q  =/=  P  /\  x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )  -> 
( x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) ) )
84 simpl3l 1060 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  ( x  e.  ( EE `  N )  /\  ( x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) ) ) )  ->  P  =/=  Q )
8584necomd 2656 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  ( x  e.  ( EE `  N )  /\  ( x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) ) ) )  ->  Q  =/=  P )
86 simprrl 772 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  ( x  e.  ( EE `  N )  /\  ( x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) ) ) )  ->  x  =/=  P )
87 simprrr 773 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  ( x  e.  ( EE `  N )  /\  ( x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) ) ) )  -> 
( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )
8885, 86, 873jca 1185 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  ( x  e.  ( EE `  N )  /\  ( x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) ) ) )  -> 
( Q  =/=  P  /\  x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) ) )
8988expr 618 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( x  =/= 
P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )  -> 
( Q  =/=  P  /\  x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) ) ) )
9083, 89impbid2 207 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( Q  =/= 
P  /\  x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )  <->  ( x  =/=  P  /\  ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) ) ) )
9182, 90bitrd 256 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( POutsideOf <. Q ,  x >.  <-> 
( x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) ) ) )
92 broutsideof2 30838 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  x  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. R ,  x >.  <-> 
( R  =/=  P  /\  x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
931, 3, 17, 2, 92syl13anc 1266 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( POutsideOf <. R ,  x >.  <-> 
( R  =/=  P  /\  x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
94 3simpc 1004 . . . . . . . . . . . 12  |-  ( ( R  =/=  P  /\  x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) )  -> 
( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
95 simpl3r 1061 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  ( x  e.  ( EE `  N )  /\  ( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )  ->  P  =/=  R )
9695necomd 2656 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  ( x  e.  ( EE `  N )  /\  ( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )  ->  R  =/=  P )
97 simprrl 772 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  ( x  e.  ( EE `  N )  /\  ( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )  ->  x  =/=  P )
98 simprrr 773 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  ( x  e.  ( EE `  N )  /\  ( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )  -> 
( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) )
9996, 97, 983jca 1185 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  ( x  e.  ( EE `  N )  /\  ( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )  -> 
( R  =/=  P  /\  x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
10099expr 618 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( x  =/= 
P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) )  -> 
( R  =/=  P  /\  x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
10194, 100impbid2 207 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( R  =/= 
P  /\  x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) )  <->  ( x  =/=  P  /\  ( R 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
10293, 101bitrd 256 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( POutsideOf <. R ,  x >.  <-> 
( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) )
10391, 102orbi12d 714 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. )  <->  ( (
x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )  \/  ( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) ) )
104103adantr 466 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( ( POutsideOf
<. Q ,  x >.  \/  POutsideOf <. R ,  x >. )  <->  ( ( x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )  \/  ( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) ) )
105104orbi2d 706 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( (
x  =  P  \/  ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. ) )  <->  ( x  =  P  \/  (
( x  =/=  P  /\  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )  \/  ( x  =/=  P  /\  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) ) ) ) )
10680, 105bitr4d 259 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( x  Colinear  <. P ,  Q >.  <->  (
x  =  P  \/  ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. ) ) ) )
107 orcom 388 . . . . . . 7  |-  ( ( x  =  P  \/  ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. ) )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  POutsideOf <. R ,  x >. )  \/  x  =  P ) )
108 or32 529 . . . . . . 7  |-  ( ( ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. )  \/  x  =  P )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) )
109107, 108bitri 252 . . . . . 6  |-  ( ( x  =  P  \/  ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. ) )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) )
110106, 109syl6bb 264 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  x  e.  ( EE `  N ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( x  Colinear  <. P ,  Q >.  <->  (
( POutsideOf <. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) ) )
111110an32s 811 . . . 4  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
)  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  P  Btwn  <. Q ,  R >. )  /\  x  e.  ( EE `  N
) )  ->  (
x  Colinear  <. P ,  Q >.  <-> 
( ( POutsideOf <. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) ) )
112111rabbidva 3012 . . 3  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  P  Btwn  <. Q ,  R >. )  ->  { x  e.  ( EE `  N
)  |  x  Colinear  <. P ,  Q >. }  =  { x  e.  ( EE `  N )  |  ( ( POutsideOf <. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) } )
113 simp1 1005 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  N  e.  NN )
114 simp21 1038 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  P  e.  ( EE `  N ) )
115 simp22 1039 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  Q  e.  ( EE `  N ) )
116 simp3l 1033 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  P  =/=  Q )
117 fvline2 30862 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  -> 
( PLine Q )  =  { x  e.  ( EE `  N
)  |  x  Colinear  <. P ,  Q >. } )
118113, 114, 115, 116, 117syl13anc 1266 . . . 4  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  -> 
( PLine Q )  =  { x  e.  ( EE `  N
)  |  x  Colinear  <. P ,  Q >. } )
119118adantr 466 . . 3  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( PLine Q )  =  {
x  e.  ( EE
`  N )  |  x  Colinear  <. P ,  Q >. } )
120 fvray 30857 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  -> 
( PRay Q )  =  { x  e.  ( EE `  N
)  |  POutsideOf <. Q ,  x >. } )
121113, 114, 115, 116, 120syl13anc 1266 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  -> 
( PRay Q )  =  { x  e.  ( EE `  N
)  |  POutsideOf <. Q ,  x >. } )
122 rabsn 4010 . . . . . . . . 9  |-  ( P  e.  ( EE `  N )  ->  { x  e.  ( EE `  N
)  |  x  =  P }  =  { P } )
123114, 122syl 17 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  { x  e.  ( EE `  N )  |  x  =  P }  =  { P } )
124123eqcomd 2434 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  { P }  =  {
x  e.  ( EE
`  N )  |  x  =  P }
)
125121, 124uneq12d 3564 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  -> 
( ( PRay Q
)  u.  { P } )  =  ( { x  e.  ( EE `  N )  |  POutsideOf <. Q ,  x >. }  u.  { x  e.  ( EE `  N
)  |  x  =  P } ) )
126 simp23 1040 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  R  e.  ( EE `  N ) )
127 simp3r 1034 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  P  =/=  R )
128 fvray 30857 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  P  =/=  R ) )  -> 
( PRay R )  =  { x  e.  ( EE `  N
)  |  POutsideOf <. R ,  x >. } )
129113, 114, 126, 127, 128syl13anc 1266 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  -> 
( PRay R )  =  { x  e.  ( EE `  N
)  |  POutsideOf <. R ,  x >. } )
130125, 129uneq12d 3564 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  -> 
( ( ( PRay Q )  u.  { P } )  u.  ( PRay R ) )  =  ( ( { x  e.  ( EE `  N
)  |  POutsideOf <. Q ,  x >. }  u.  {
x  e.  ( EE
`  N )  |  x  =  P }
)  u.  { x  e.  ( EE `  N
)  |  POutsideOf <. R ,  x >. } ) )
131130adantr 466 . . . 4  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( (
( PRay Q )  u.  { P }
)  u.  ( PRay R ) )  =  ( ( { x  e.  ( EE `  N
)  |  POutsideOf <. Q ,  x >. }  u.  {
x  e.  ( EE
`  N )  |  x  =  P }
)  u.  { x  e.  ( EE `  N
)  |  POutsideOf <. R ,  x >. } ) )
132 unrab 3687 . . . . . 6  |-  ( { x  e.  ( EE
`  N )  |  POutsideOf <. Q ,  x >. }  u.  { x  e.  ( EE `  N
)  |  x  =  P } )  =  { x  e.  ( EE `  N )  |  ( POutsideOf <. Q ,  x >.  \/  x  =  P ) }
133132uneq1i 3559 . . . . 5  |-  ( ( { x  e.  ( EE `  N )  |  POutsideOf <. Q ,  x >. }  u.  { x  e.  ( EE `  N
)  |  x  =  P } )  u. 
{ x  e.  ( EE `  N )  |  POutsideOf <. R ,  x >. } )  =  ( { x  e.  ( EE `  N )  |  ( POutsideOf <. Q ,  x >.  \/  x  =  P ) }  u.  { x  e.  ( EE
`  N )  |  POutsideOf <. R ,  x >. } )
134 unrab 3687 . . . . 5  |-  ( { x  e.  ( EE
`  N )  |  ( POutsideOf <. Q ,  x >.  \/  x  =  P ) }  u.  {
x  e.  ( EE
`  N )  |  POutsideOf <. R ,  x >. } )  =  {
x  e.  ( EE
`  N )  |  ( ( POutsideOf <. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) }
135133, 134eqtri 2450 . . . 4  |-  ( ( { x  e.  ( EE `  N )  |  POutsideOf <. Q ,  x >. }  u.  { x  e.  ( EE `  N
)  |  x  =  P } )  u. 
{ x  e.  ( EE `  N )  |  POutsideOf <. R ,  x >. } )  =  {
x  e.  ( EE
`  N )  |  ( ( POutsideOf <. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) }
136131, 135syl6eq 2478 . . 3  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( (
( PRay Q )  u.  { P }
)  u.  ( PRay R ) )  =  { x  e.  ( EE `  N )  |  ( ( POutsideOf <. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) } )
137112, 119, 1363eqtr4d 2472 . 2  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  /\  P  Btwn  <. Q ,  R >. )  ->  ( PLine Q )  =  ( ( ( PRay Q
)  u.  { P } )  u.  ( PRay R ) ) )
138137ex 435 1  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  -> 
( P  Btwn  <. Q ,  R >.  ->  ( PLine Q )  =  ( ( ( PRay Q
)  u.  { P } )  u.  ( PRay R ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   {crab 2718    u. cun 3377   {csn 3941   <.cop 3947   class class class wbr 4366   ` cfv 5544  (class class class)co 6249   NNcn 10560   EEcee 24860    Btwn cbtwn 24861    Colinear ccolin 30753  OutsideOfcoutsideof 30835  Linecline2 30850  Raycray 30851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-inf2 8099  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-ec 7320  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-sup 7909  df-oi 7978  df-card 8325  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-n0 10821  df-z 10889  df-uz 11111  df-rp 11254  df-ico 11592  df-icc 11593  df-fz 11736  df-fzo 11867  df-seq 12164  df-exp 12223  df-hash 12466  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-clim 13495  df-sum 13696  df-ee 24863  df-btwn 24864  df-cgr 24865  df-ofs 30699  df-colinear 30755  df-ifs 30756  df-cgr3 30757  df-fs 30758  df-outsideof 30836  df-line2 30853  df-ray 30854
This theorem is referenced by: (None)
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