Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lineunray Unicode version

Theorem lineunray 24177
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
StepHypRef Expression
1 simpl1 963 . . . . . . . . . . . . 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 449 . . . . . . . . . . . . 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 1038 . . . . . . . . . . . . 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 1039 . . . . . . . . . . . . 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 24089 . . . . . . . . . . . . 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 1189 . . . . . . . . . . . 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 453 . . . . . . . . . . 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 375 . . . . . . . . . . . . . 14  |-  ( x 
Btwn  <. P ,  Q >.  ->  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )
98orcd 383 . . . . . . . . . . . . 13  |-  ( x 
Btwn  <. P ,  Q >.  ->  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
109a1i 12 . . . . . . . . . . . 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 1015 . . . . . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . . . . 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 1137 . . . . . . . . . . . . . . 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 1040 . . . . . . . . . . . . . . . . 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 24132 . . . . . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . 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 384 . . . . . . . . . . . . 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 601 . . . . . . . . . . . 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 24044 . . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . . 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 376 . . . . . . . . . . . . . . 15  |-  ( Q 
Btwn  <. P ,  x >.  ->  ( Q  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. ) )
2726orcd 383 . . . . . . . . . . . . . 14  |-  ( Q 
Btwn  <. P ,  x >.  ->  ( ( Q 
Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  Q >. )  \/  ( R  Btwn  <. P ,  x >.  \/  x  Btwn  <. P ,  R >. ) ) )
2825, 27syl6bi 221 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 1251 . . . . . . . . . . 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 208 . . . . . . . . . 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 375 . . . . . . . . . 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 24097 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) )  ->  P  Colinear  <. P ,  Q >. )
351, 3, 4, 34syl3anc 1187 . . . . . . . . . . . 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 4027 . . . . . . . . . . . 12  |-  ( x  =  P  ->  (
x  Colinear  <. P ,  Q >.  <-> 
P  Colinear  <. P ,  Q >. ) )
3735, 36syl5ibrcom 215 . . . . . . . . . . 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 453 . . . . . . . . . 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 24101 . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . 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 24103 . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . 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 371 . . . . . . . . . . . 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 453 . . . . . . . . . . 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 1016 . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . . . . 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 1137 . . . . . . . . . . . . . . 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 24054 . . . . . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . 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 24102 . . . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . 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 601 . . . . . . . . . . . 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 736 . . . . . . . . . . . . . . . 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 24045 . . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . 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 24045 . . . . . . . . . . . . . . 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 24051 . . . . . . . . . . . . . 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 24100 . . . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . 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 601 . . . . . . . . . . . 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 371 . . . . . . . . . . 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 371 . . . . . . . . . 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 371 . . . . . . . . 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 185 . . . . . . . 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 pm5.63 895 . . . . . . . . 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 2449 . . . . . . . . . . . 12  |-  ( x  =/=  P  <->  -.  x  =  P )
7574anbi1i 679 . . . . . . . . . . 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 842 . . . . . . . . . . 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 244 . . . . . . . . . 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 507 . . . . . . . . 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 242 . . . . . . . 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 254 . . . . . . 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 24152 . . . . . . . . . . . 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 1189 . . . . . . . . . . 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 959 . . . . . . . . . . . 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 1015 . . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . 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 743 . . . . . . . . . . . . . 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 744 . . . . . . . . . . . . . 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 1137 . . . . . . . . . . . . 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 601 . . . . . . . . . . . 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 197 . . . . . . . . . . 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 246 . . . . . . . . . 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 24152 . . . . . . . . . . . 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 1189 . . . . . . . . . . 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 959 . . . . . . . . . . . 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 1016 . . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . 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 743 . . . . . . . . . . . . . 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 744 . . . . . . . . . . . . . 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 1137 . . . . . . . . . . . . 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 601 . . . . . . . . . . . 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 197 . . . . . . . . . . 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 246 . . . . . . . . . 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 693 . . . . . . . . 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 453 . . . . . . . 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 685 . . . . . . 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 249 . . . . . 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 378 . . . . . . 7  |-  ( ( x  =  P  \/  ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. ) )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  POutsideOf <. R ,  x >. )  \/  x  =  P ) )
108 or32 515 . . . . . . 7  |-  ( ( ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. )  \/  x  =  P )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) )
109107, 108bitri 242 . . . . . 6  |-  ( ( x  =  P  \/  ( POutsideOf <. Q ,  x >.  \/  POutsideOf <. R ,  x >. ) )  <->  ( ( POutsideOf
<. Q ,  x >.  \/  x  =  P )  \/  POutsideOf <. R ,  x >. ) )
110106, 109syl6bb 254 . . . . 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 782 . . . 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 2780 . . 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 960 . . . . 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 993 . . . . 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 994 . . . . 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 988 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  P  =/=  Q )
117 fvline2 24176 . . . . 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 1189 . . . 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 453 . . 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 24171 . . . . . . . 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 1189 . . . . . . 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 3698 . . . . . . . . 9  |-  ( P  e.  ( EE `  N )  ->  { x  e.  ( EE `  N
)  |  x  =  P }  =  { P } )
123114, 122syl 17 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  { x  e.  ( EE `  N )  |  x  =  P }  =  { P } )
124123eqcomd 2289 . . . . . . 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 3331 . . . . . 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 995 . . . . . . 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 989 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( P  =/=  Q  /\  P  =/=  R ) )  ->  P  =/=  R )
128 fvray 24171 . . . . . . 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 1189 . . . . . 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 3331 . . . . 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 453 . . . 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 3440 . . . . . 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 3326 . . . . 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 3440 . . . . 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 2304 . . . 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 2332 . . 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 2326 . 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 425 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 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    \/ w3o 938    /\ w3a 939    = wceq 1628    e. wcel 1688    =/= wne 2447   {crab 2548    u. cun 3151   {csn 3641   <.cop 3644   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   NNcn 9741   EEcee 23923    Btwn cbtwn 23924    Colinear ccolin 24067  OutsideOfcoutsideof 24149  Linecline2 24164  Raycray 24165
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-ec 6657  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-sup 7189  df-oi 7220  df-card 7567  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-seq 11041  df-exp 11099  df-hash 11332  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-clim 11956  df-sum 12153  df-ee 23926  df-btwn 23927  df-cgr 23928  df-ofs 24013  df-ifs 24069  df-cgr3 24070  df-colinear 24071  df-fs 24072  df-outsideof 24150  df-line2 24167  df-ray 24168
  Copyright terms: Public domain W3C validator