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Theorem lineunray 28345
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 991 . . . . . . . . . . . . 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 461 . . . . . . . . . . . . 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 1066 . . . . . . . . . . . . 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 1067 . . . . . . . . . . . . 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 28257 . . . . . . . . . . . . 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 1221 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 384 . . . . . . . . . . . . . 14  |-  ( x 
Btwn  <. P ,  Q >.  ->  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )
98orcd 392 . . . . . . . . . . . . 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 1043 . . . . . . . . . . . . . . . . . 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 2723 . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . . . . 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 1068 . . . . . . . . . . . . . . . . 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 28300 . . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . 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 393 . . . . . . . . . . . . 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 615 . . . . . . . . . . . 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 28212 . . . . . . . . . . . . . . 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 1221 . . . . . . . . . . . . . 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 385 . . . . . . . . . . . . . . 15  |-  ( Q 
Btwn  <. P ,  x >.  ->  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )
2726orcd 392 . . . . . . . . . . . . . 14  |-  ( Q 
Btwn  <. P ,  x >.  ->  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
2825, 27syl6bi 228 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 1283 . . . . . . . . . . 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 215 . . . . . . . . . 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 384 . . . . . . . . . 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 33 . . . . . . . . 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 28265 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) )  ->  P  Colinear  <. P ,  Q >. )
351, 3, 4, 34syl3anc 1219 . . . . . . . . . . . 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 4406 . . . . . . . . . . . 12  |-  ( x  =  P  ->  (
x  Colinear  <. P ,  Q >.  <-> 
P  Colinear  <. P ,  Q >. ) )
3735, 36syl5ibrcom 222 . . . . . . . . . . 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 465 . . . . . . . . . 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 28269 . . . . . . . . . . . . . 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 1221 . . . . . . . . . . . . 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 28271 . . . . . . . . . . . . . 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 1221 . . . . . . . . . . . . 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 380 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 1044 . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . . . . 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 28222 . . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . 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 28270 . . . . . . . . . . . . . . . 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 1221 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 615 . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . 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 28213 . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . 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 28213 . . . . . . . . . . . . . . 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 28219 . . . . . . . . . . . . . 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 28268 . . . . . . . . . . . . . . . 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 1221 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 615 . . . . . . . . . . . 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 380 . . . . . . . . . . 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 380 . . . . . . . . . 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 380 . . . . . . . . 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 191 . . . . . . . 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 pm5.63 915 . . . . . . . . 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 2650 . . . . . . . . . . . 12  |-  ( x  =/=  P  <->  -.  x  =  P )
7574anbi1i 695 . . . . . . . . . . 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 862 . . . . . . . . . . 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 251 . . . . . . . . . 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 519 . . . . . . . . 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 249 . . . . . . . 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 261 . . . . . . 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 28320 . . . . . . . . . . . 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 1221 . . . . . . . . . . 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 987 . . . . . . . . . . . 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 1043 . . . . . . . . . . . . . . 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 2723 . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . . 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 615 . . . . . . . . . . . 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 204 . . . . . . . . . . 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 253 . . . . . . . . . 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 28320 . . . . . . . . . . . 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 1221 . . . . . . . . . . 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 987 . . . . . . . . . . . 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 1044 . . . . . . . . . . . . . . 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 2723 . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . . 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 615 . . . . . . . . . . . 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 204 . . . . . . . . . . 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 253 . . . . . . . . . 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 709 . . . . . . . . 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 465 . . . . . . . 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 701 . . . . . . 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 256 . . . . . 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 387 . . . . . . 7  |-  ( ( x  =  P  \/  ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. ) )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  POutsideOf <. R ,  x >. )  \/  x  =  P ) )
108 or32 527 . . . . . . 7  |-  ( ( ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. )  \/  x  =  P )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) )
109107, 108bitri 249 . . . . . 6  |-  ( ( x  =  P  \/  ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. ) )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) )
110106, 109syl6bb 261 . . . . 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 802 . . . 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 3069 . . 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 988 . . . . 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 1021 . . . . 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 1022 . . . . 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 1016 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  P  =/=  Q )
117 fvline2 28344 . . . . 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 1221 . . . 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 465 . . 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 28339 . . . . . . . 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 1221 . . . . . . 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 4053 . . . . . . . . 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 2462 . . . . . . 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 3622 . . . . . 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 1023 . . . . . . 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 1017 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  P  =/=  R )
128 fvray 28339 . . . . . . 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 1221 . . . . . 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 3622 . . . . 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 465 . . . 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 3732 . . . . . 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 3617 . . . . 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 3732 . . . . 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 2483 . . . 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 2511 . . 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 2505 . 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 434 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 184    \/ wo 368    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   {crab 2803    u. cun 3437   {csn 3988   <.cop 3994   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   NNcn 10437   EEcee 23313    Btwn cbtwn 23314    Colinear ccolin 28235  OutsideOfcoutsideof 28317  Linecline2 28332  Raycray 28333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-ec 7216  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-oi 7839  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-ico 11421  df-icc 11422  df-fz 11559  df-fzo 11670  df-seq 11928  df-exp 11987  df-hash 12225  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088  df-sum 13286  df-ee 23316  df-btwn 23317  df-cgr 23318  df-ofs 28181  df-colinear 28237  df-ifs 28238  df-cgr3 28239  df-fs 28240  df-outsideof 28318  df-line2 28335  df-ray 28336
This theorem is referenced by: (None)
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