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Theorem lineunray 25788
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 960 . . . . . . . . . . . . 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 448 . . . . . . . . . . . . 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 1035 . . . . . . . . . . . . 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 1036 . . . . . . . . . . . . 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 25700 . . . . . . . . . . . . 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 1186 . . . . . . . . . . . 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 452 . . . . . . . . . . 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 374 . . . . . . . . . . . . . 14  |-  ( x 
Btwn  <. P ,  Q >.  ->  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )
98orcd 382 . . . . . . . . . . . . 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 1012 . . . . . . . . . . . . . . . . . 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 2626 . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . . 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 1037 . . . . . . . . . . . . . . . . 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 25743 . . . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . 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 383 . . . . . . . . . . . . 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 599 . . . . . . . . . . . 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 25655 . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . 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 375 . . . . . . . . . . . . . . 15  |-  ( Q 
Btwn  <. P ,  x >.  ->  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )
2726orcd 382 . . . . . . . . . . . . . 14  |-  ( Q 
Btwn  <. P ,  x >.  ->  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
2825, 27syl6bi 220 . . . . . . . . . . . . 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 452 . . . . . . . . . . . 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 1248 . . . . . . . . . . 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 207 . . . . . . . . . 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 374 . . . . . . . . . 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 31 . . . . . . . . 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 25708 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) )  ->  P  Colinear  <. P ,  Q >. )
351, 3, 4, 34syl3anc 1184 . . . . . . . . . . . 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 4149 . . . . . . . . . . . 12  |-  ( x  =  P  ->  (
x  Colinear  <. P ,  Q >.  <-> 
P  Colinear  <. P ,  Q >. ) )
3735, 36syl5ibrcom 214 . . . . . . . . . . 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 452 . . . . . . . . . 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 25712 . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . 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 25714 . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . 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 370 . . . . . . . . . . . 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 452 . . . . . . . . . . 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 1013 . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . . 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 25665 . . . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . 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 25713 . . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . 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 599 . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . 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 25656 . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . 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 25656 . . . . . . . . . . . . . . 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 25662 . . . . . . . . . . . . . 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 25711 . . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . 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 599 . . . . . . . . . . . 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 370 . . . . . . . . . . 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 370 . . . . . . . . . 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 370 . . . . . . . . 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 184 . . . . . . . 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 pm5.63 891 . . . . . . . . 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 2545 . . . . . . . . . . . 12  |-  ( x  =/=  P  <->  -.  x  =  P )
7574anbi1i 677 . . . . . . . . . . 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 838 . . . . . . . . . . 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 243 . . . . . . . . . 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 506 . . . . . . . . 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 241 . . . . . . . 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 253 . . . . . . 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 25763 . . . . . . . . . . . 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 1186 . . . . . . . . . . 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 956 . . . . . . . . . . . 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 1012 . . . . . . . . . . . . . . 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 2626 . . . . . . . . . . . . . 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 741 . . . . . . . . . . . . . 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 742 . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . 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 599 . . . . . . . . . . . 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 196 . . . . . . . . . . 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 245 . . . . . . . . . 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 25763 . . . . . . . . . . . 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 1186 . . . . . . . . . . 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 956 . . . . . . . . . . . 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 1013 . . . . . . . . . . . . . . 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 2626 . . . . . . . . . . . . . 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 741 . . . . . . . . . . . . . 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 742 . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . 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 599 . . . . . . . . . . . 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 196 . . . . . . . . . . 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 245 . . . . . . . . . 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 691 . . . . . . . . 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 452 . . . . . . . 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 683 . . . . . . 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 248 . . . . . 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 377 . . . . . . 7  |-  ( ( x  =  P  \/  ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. ) )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  POutsideOf <. R ,  x >. )  \/  x  =  P ) )
108 or32 514 . . . . . . 7  |-  ( ( ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. )  \/  x  =  P )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) )
109107, 108bitri 241 . . . . . 6  |-  ( ( x  =  P  \/  ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. ) )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) )
110106, 109syl6bb 253 . . . . 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 780 . . . 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 2883 . . 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 957 . . . . 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 990 . . . . 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 991 . . . . 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 985 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  P  =/=  Q )
117 fvline2 25787 . . . . 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 1186 . . . 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 452 . . 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 25782 . . . . . . . 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 1186 . . . . . . 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 3809 . . . . . . . . 9  |-  ( P  e.  ( EE `  N )  ->  { x  e.  ( EE `  N
)  |  x  =  P }  =  { P } )
123114, 122syl 16 . . . . . . . 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 2385 . . . . . . 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 3438 . . . . . 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 992 . . . . . . 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 986 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  P  =/=  R )
128 fvray 25782 . . . . . . 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 1186 . . . . . 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 3438 . . . . 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 452 . . . 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 3548 . . . . . 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 3433 . . . . 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 3548 . . . . 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 2400 . . . 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 2428 . . 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 2422 . 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 424 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   {crab 2646    u. cun 3254   {csn 3750   <.cop 3753   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   NNcn 9925   EEcee 25534    Btwn cbtwn 25535    Colinear ccolin 25678  OutsideOfcoutsideof 25760  Linecline2 25775  Raycray 25776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-ec 6836  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-ico 10847  df-icc 10848  df-fz 10969  df-fzo 11059  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-sum 12400  df-ee 25537  df-btwn 25538  df-cgr 25539  df-ofs 25624  df-ifs 25680  df-cgr3 25681  df-colinear 25682  df-fs 25683  df-outsideof 25761  df-line2 25778  df-ray 25779
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