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Theorem lineunray 24770
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 958 . . . . . . . . . . . . 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 447 . . . . . . . . . . . . 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 1033 . . . . . . . . . . . . 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 1034 . . . . . . . . . . . . 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 24682 . . . . . . . . . . . . 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 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 ) )  -> 
( x  Colinear  <. P ,  Q >. 
<->  ( x  Btwn  <. P ,  Q >.  \/  P  Btwn  <. Q ,  x >.  \/  Q  Btwn  <. x ,  P >. ) ) )
76adantr 451 . . . . . . . . . . 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 373 . . . . . . . . . . . . . 14  |-  ( x 
Btwn  <. P ,  Q >.  ->  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )
98orcd 381 . . . . . . . . . . . . 13  |-  ( x 
Btwn  <. P ,  Q >.  ->  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
109a1i 10 . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . . . . 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 2529 . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . 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 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 ,  x >. )
1613, 14, 153jca 1132 . . . . . . . . . . . . . . 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 1035 . . . . . . . . . . . . . . . . 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 24725 . . . . . . . . . . . . . . . . 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 1191 . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . 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 14 . . . . . . . . . . . . . 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 382 . . . . . . . . . . . . 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 598 . . . . . . . . . . . 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 24637 . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . 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 374 . . . . . . . . . . . . . . 15  |-  ( Q 
Btwn  <. P ,  x >.  ->  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )
2726orcd 381 . . . . . . . . . . . . . 14  |-  ( Q 
Btwn  <. P ,  x >.  ->  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
2825, 27syl6bi 219 . . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 1246 . . . . . . . . . . 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 206 . . . . . . . . . 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 373 . . . . . . . . . 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 29 . . . . . . . . 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 24690 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) )  ->  P  Colinear  <. P ,  Q >. )
351, 3, 4, 34syl3anc 1182 . . . . . . . . . . . 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 4026 . . . . . . . . . . . 12  |-  ( x  =  P  ->  (
x  Colinear  <. P ,  Q >.  <-> 
P  Colinear  <. P ,  Q >. ) )
3735, 36syl5ibrcom 213 . . . . . . . . . . 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 451 . . . . . . . . . 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 24694 . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 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 24696 . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 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 369 . . . . . . . . . . . 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 451 . . . . . . . . . . 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 1011 . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . 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 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 >. ) )  ->  R  Btwn  <. P ,  x >. )
4946, 47, 483jca 1132 . . . . . . . . . . . . . . 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 24647 . . . . . . . . . . . . . . . . 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 1191 . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . 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 14 . . . . . . . . . . . . . 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 24695 . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . 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 14 . . . . . . . . . . . . 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 598 . . . . . . . . . . . 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 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 >. ) )  ->  x  Btwn  <. P ,  R >. )
601, 2, 3, 17, 59btwncomand 24638 . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . 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 24638 . . . . . . . . . . . . . . 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 24644 . . . . . . . . . . . . . 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 24693 . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . 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 14 . . . . . . . . . . . . 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 598 . . . . . . . . . . . 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 369 . . . . . . . . . . 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 369 . . . . . . . . . 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 369 . . . . . . . . 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 183 . . . . . . . 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 pm5.63 890 . . . . . . . . 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 2448 . . . . . . . . . . . 12  |-  ( x  =/=  P  <->  -.  x  =  P )
7574anbi1i 676 . . . . . . . . . . 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 837 . . . . . . . . . . 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 242 . . . . . . . . . 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 505 . . . . . . . . 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 240 . . . . . . . 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 252 . . . . . . 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 24745 . . . . . . . . . . . 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 1184 . . . . . . . . . . 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 954 . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . 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 2529 . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . 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 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 >. ) ) ) )  -> 
( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )
8885, 86, 873jca 1132 . . . . . . . . . . . . 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 598 . . . . . . . . . . . 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 195 . . . . . . . . . . 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 244 . . . . . . . . . 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 24745 . . . . . . . . . . . 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 1184 . . . . . . . . . . 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 954 . . . . . . . . . . . 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 1011 . . . . . . . . . . . . . . 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 2529 . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . 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 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 >. ) ) ) )  -> 
( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) )
9996, 97, 983jca 1132 . . . . . . . . . . . . 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 598 . . . . . . . . . . . 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 195 . . . . . . . . . . 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 244 . . . . . . . . . 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 690 . . . . . . . . 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 451 . . . . . . . 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 682 . . . . . . 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 247 . . . . . 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 376 . . . . . . 7  |-  ( ( x  =  P  \/  ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. ) )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  POutsideOf <. R ,  x >. )  \/  x  =  P ) )
108 or32 513 . . . . . . 7  |-  ( ( ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. )  \/  x  =  P )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) )
109107, 108bitri 240 . . . . . 6  |-  ( ( x  =  P  \/  ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. ) )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) )
110106, 109syl6bb 252 . . . . 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 779 . . . 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 2779 . . 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 955 . . . . 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 988 . . . . 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 989 . . . . 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 983 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  P  =/=  Q )
117 fvline2 24769 . . . . 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 1184 . . . 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 451 . . 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 24764 . . . . . . . 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 1184 . . . . . . 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 3697 . . . . . . . . 9  |-  ( P  e.  ( EE `  N )  ->  { x  e.  ( EE `  N
)  |  x  =  P }  =  { P } )
123114, 122syl 15 . . . . . . . 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 2288 . . . . . . 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 3330 . . . . . 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 990 . . . . . . 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 984 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  P  =/=  R )
128 fvray 24764 . . . . . . 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 1184 . . . . . 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 3330 . . . . 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 451 . . . 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 3439 . . . . . 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 3325 . . . . 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 3439 . . . . 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 2303 . . . 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 2331 . . 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 2325 . 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 423 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 176    \/ wo 357    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547    u. cun 3150   {csn 3640   <.cop 3643   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   NNcn 9746   EEcee 24516    Btwn cbtwn 24517    Colinear ccolin 24660  OutsideOfcoutsideof 24742  Linecline2 24757  Raycray 24758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ec 6662  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-ee 24519  df-btwn 24520  df-cgr 24521  df-ofs 24606  df-ifs 24662  df-cgr3 24663  df-colinear 24664  df-fs 24665  df-outsideof 24743  df-line2 24760  df-ray 24761
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