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

Theorem outsideoftr 25971
Description: Transitivity law for outsideness. Theorem 6.7 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
outsideoftr  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( ( POutsideOf <. A ,  B >.  /\  POutsideOf <. B ,  C >. )  ->  POutsideOf <. A ,  C >. ) )

Proof of Theorem outsideoftr
StepHypRef Expression
1 simpll 731 . . . . 5  |-  ( ( ( A  =/=  P  /\  B  =/=  P
)  /\  ( B  =/=  P  /\  C  =/= 
P ) )  ->  A  =/=  P )
2 simplr 732 . . . . 5  |-  ( ( ( A  =/=  P  /\  B  =/=  P
)  /\  ( B  =/=  P  /\  C  =/= 
P ) )  ->  B  =/=  P )
3 simprr 734 . . . . 5  |-  ( ( ( A  =/=  P  /\  B  =/=  P
)  /\  ( B  =/=  P  /\  C  =/= 
P ) )  ->  C  =/=  P )
41, 2, 33jca 1134 . . . 4  |-  ( ( ( A  =/=  P  /\  B  =/=  P
)  /\  ( B  =/=  P  /\  C  =/= 
P ) )  -> 
( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
) )
5 simplr1 999 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
) )  /\  (
( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) )  ->  A  =/=  P
)
6 simplr3 1001 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
) )  /\  (
( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) )  ->  C  =/=  P
)
7 df-3an 938 . . . . . . . . . . . 12  |-  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
)  /\  A  Btwn  <. P ,  B >.  /\  B  Btwn  <. P ,  C >. )  <->  ( (
( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
)  /\  A  Btwn  <. P ,  B >. )  /\  B  Btwn  <. P ,  C >. ) )
8 simp1 957 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  N  e.  NN )
9 simp3r 986 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  P  e.  ( EE `  N ) )
10 simp2l 983 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N ) )
11 simp2r 984 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N ) )
12 simp3l 985 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N ) )
13 simpr2 964 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  A  Btwn  <. P ,  B >.  /\  B  Btwn  <. P ,  C >. ) )  ->  A  Btwn  <. P ,  B >. )
14 simpr3 965 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  A  Btwn  <. P ,  B >.  /\  B  Btwn  <. P ,  C >. ) )  ->  B  Btwn  <. P ,  C >. )
158, 9, 10, 11, 12, 13, 14btwnexchand 25868 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  A  Btwn  <. P ,  B >.  /\  B  Btwn  <. P ,  C >. ) )  ->  A  Btwn  <. P ,  C >. )
1615orcd 382 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  A  Btwn  <. P ,  B >.  /\  B  Btwn  <. P ,  C >. ) )  -> 
( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) )
177, 16sylan2br 463 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  A  Btwn  <. P ,  B >. )  /\  B  Btwn  <. P ,  C >. ) )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) )
1817expr 599 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  A  Btwn  <. P ,  B >. ) )  ->  ( B  Btwn  <. P ,  C >.  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) )
19 simprlr 740 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  A  Btwn  <. P ,  B >. )  /\  C  Btwn  <. P ,  B >. ) )  ->  A  Btwn  <. P ,  B >. )
20 simprr 734 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  A  Btwn  <. P ,  B >. )  /\  C  Btwn  <. P ,  B >. ) )  ->  C  Btwn  <. P ,  B >. )
21 btwnconn3 25945 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( ( A 
Btwn  <. P ,  B >.  /\  C  Btwn  <. P ,  B >. )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) )
228, 9, 10, 12, 11, 21syl122anc 1193 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( ( A 
Btwn  <. P ,  B >.  /\  C  Btwn  <. P ,  B >. )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) )
2322adantr 452 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  A  Btwn  <. P ,  B >. )  /\  C  Btwn  <. P ,  B >. ) )  ->  ( ( A  Btwn  <. P ,  B >.  /\  C  Btwn  <. P ,  B >. )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) )
2419, 20, 23mp2and 661 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  A  Btwn  <. P ,  B >. )  /\  C  Btwn  <. P ,  B >. ) )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) )
2524expr 599 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  A  Btwn  <. P ,  B >. ) )  ->  ( C  Btwn  <. P ,  B >.  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) )
2618, 25jaod 370 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  A  Btwn  <. P ,  B >. ) )  ->  ( ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) )
2726expr 599 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
) )  ->  ( A  Btwn  <. P ,  B >.  ->  ( ( B 
Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) ) )
28 simpll2 997 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  B  Btwn  <. P ,  A >. )  /\  B  Btwn  <. P ,  C >. )  ->  B  =/=  P )
2928adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  B  Btwn  <. P ,  A >. )  /\  B  Btwn  <. P ,  C >. ) )  ->  B  =/=  P )
3029necomd 2654 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  B  Btwn  <. P ,  A >. )  /\  B  Btwn  <. P ,  C >. ) )  ->  P  =/=  B )
31 simprlr 740 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  B  Btwn  <. P ,  A >. )  /\  B  Btwn  <. P ,  C >. ) )  ->  B  Btwn  <. P ,  A >. )
32 simprr 734 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  B  Btwn  <. P ,  A >. )  /\  B  Btwn  <. P ,  C >. ) )  ->  B  Btwn  <. P ,  C >. )
33 btwnconn1 25943 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( ( P  =/=  B  /\  B  Btwn  <. P ,  A >.  /\  B  Btwn  <. P ,  C >. )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) )
348, 9, 11, 10, 12, 33syl122anc 1193 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( ( P  =/=  B  /\  B  Btwn  <. P ,  A >.  /\  B  Btwn  <. P ,  C >. )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) )
3534adantr 452 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  B  Btwn  <. P ,  A >. )  /\  B  Btwn  <. P ,  C >. ) )  ->  ( ( P  =/=  B  /\  B  Btwn  <. P ,  A >.  /\  B  Btwn  <. P ,  C >. )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) )
3630, 31, 32, 35mp3and 1282 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  B  Btwn  <. P ,  A >. )  /\  B  Btwn  <. P ,  C >. ) )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) )
3736expr 599 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  B  Btwn  <. P ,  A >. ) )  ->  ( B  Btwn  <. P ,  C >.  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) )
38 df-3an 938 . . . . . . . . . . . 12  |-  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
)  /\  B  Btwn  <. P ,  A >.  /\  C  Btwn  <. P ,  B >. )  <->  ( (
( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
)  /\  B  Btwn  <. P ,  A >. )  /\  C  Btwn  <. P ,  B >. ) )
39 simpr3 965 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  B  Btwn  <. P ,  A >.  /\  C  Btwn  <. P ,  B >. ) )  ->  C  Btwn  <. P ,  B >. )
40 simpr2 964 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  B  Btwn  <. P ,  A >.  /\  C  Btwn  <. P ,  B >. ) )  ->  B  Btwn  <. P ,  A >. )
418, 9, 12, 11, 10, 39, 40btwnexchand 25868 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  B  Btwn  <. P ,  A >.  /\  C  Btwn  <. P ,  B >. ) )  ->  C  Btwn  <. P ,  A >. )
4241olcd 383 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  B  Btwn  <. P ,  A >.  /\  C  Btwn  <. P ,  B >. ) )  -> 
( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) )
4338, 42sylan2br 463 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  B  Btwn  <. P ,  A >. )  /\  C  Btwn  <. P ,  B >. ) )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) )
4443expr 599 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  B  Btwn  <. P ,  A >. ) )  ->  ( C  Btwn  <. P ,  B >.  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) )
4537, 44jaod 370 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  B  Btwn  <. P ,  A >. ) )  ->  ( ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) )
4645expr 599 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
) )  ->  ( B  Btwn  <. P ,  A >.  ->  ( ( B 
Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) ) )
4727, 46jaod 370 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
) )  ->  (
( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  ->  ( ( B 
Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) ) )
4847imp32 423 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
) )  /\  (
( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) )  ->  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) )
495, 6, 483jca 1134 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
) )  /\  (
( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) )  ->  ( A  =/= 
P  /\  C  =/=  P  /\  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) )
5049exp31 588 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  ->  (
( ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) )  -> 
( A  =/=  P  /\  C  =/=  P  /\  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) ) ) )
514, 50syl5 30 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( ( ( A  =/=  P  /\  B  =/=  P )  /\  ( B  =/=  P  /\  C  =/=  P
) )  ->  (
( ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) )  -> 
( A  =/=  P  /\  C  =/=  P  /\  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) ) ) )
5251imp3a 421 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( ( ( ( A  =/=  P  /\  B  =/=  P
)  /\  ( B  =/=  P  /\  C  =/= 
P ) )  /\  ( ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) )  ->  ( A  =/= 
P  /\  C  =/=  P  /\  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) ) )
53 broutsideof2 25964 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. A ,  B >.  <-> 
( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
548, 9, 10, 11, 53syl13anc 1186 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >. 
<->  ( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
55 broutsideof2 25964 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. B ,  C >.  <-> 
( B  =/=  P  /\  C  =/=  P  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) ) )
568, 9, 11, 12, 55syl13anc 1186 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. B ,  C >. 
<->  ( B  =/=  P  /\  C  =/=  P  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) ) )
5754, 56anbi12d 692 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( ( POutsideOf <. A ,  B >.  /\  POutsideOf <. B ,  C >. )  <->  ( ( A  =/=  P  /\  B  =/=  P  /\  ( A 
Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  /\  ( B  =/=  P  /\  C  =/=  P  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) ) ) )
58 df-3an 938 . . . . 5  |-  ( ( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  <->  ( ( A  =/=  P  /\  B  =/=  P )  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
59 df-3an 938 . . . . 5  |-  ( ( B  =/=  P  /\  C  =/=  P  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) )  <->  ( ( B  =/=  P  /\  C  =/=  P )  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) )
6058, 59anbi12i 679 . . . 4  |-  ( ( ( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  /\  ( B  =/=  P  /\  C  =/=  P  /\  ( B 
Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) )  <-> 
( ( ( A  =/=  P  /\  B  =/=  P )  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  /\  ( ( B  =/= 
P  /\  C  =/=  P )  /\  ( B 
Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) ) )
61 an4 798 . . . 4  |-  ( ( ( ( A  =/= 
P  /\  B  =/=  P )  /\  ( B  =/=  P  /\  C  =/=  P ) )  /\  ( ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) )  <-> 
( ( ( A  =/=  P  /\  B  =/=  P )  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  /\  ( ( B  =/= 
P  /\  C  =/=  P )  /\  ( B 
Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) ) )
6260, 61bitr4i 244 . . 3  |-  ( ( ( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  /\  ( B  =/=  P  /\  C  =/=  P  /\  ( B 
Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) )  <-> 
( ( ( A  =/=  P  /\  B  =/=  P )  /\  ( B  =/=  P  /\  C  =/=  P ) )  /\  ( ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) ) )
6357, 62syl6bb 253 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( ( POutsideOf <. A ,  B >.  /\  POutsideOf <. B ,  C >. )  <->  ( ( ( A  =/=  P  /\  B  =/=  P )  /\  ( B  =/=  P  /\  C  =/=  P
) )  /\  (
( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  /\  ( B  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  B >. ) ) ) ) )
64 broutsideof2 25964 . . 3  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. A ,  C >.  <-> 
( A  =/=  P  /\  C  =/=  P  /\  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) ) )
658, 9, 10, 12, 64syl13anc 1186 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  C >. 
<->  ( A  =/=  P  /\  C  =/=  P  /\  ( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) ) ) )
6652, 63, 653imtr4d 260 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( ( POutsideOf <. A ,  B >.  /\  POutsideOf <. B ,  C >. )  ->  POutsideOf <. A ,  C >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    e. wcel 1721    =/= wne 2571   <.cop 3781   class class class wbr 4176   ` cfv 5417   NNcn 9960   EEcee 25735    Btwn cbtwn 25736  OutsideOfcoutsideof 25961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-ico 10882  df-icc 10883  df-fz 11004  df-fzo 11095  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-sum 12439  df-ee 25738  df-btwn 25739  df-cgr 25740  df-ofs 25825  df-ifs 25881  df-cgr3 25882  df-colinear 25883  df-fs 25884  df-outsideof 25962
  Copyright terms: Public domain W3C validator