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Theorem outsideoftr 24092
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 733 . . . . 5  |-  ( ( ( A  =/=  P  /\  B  =/=  P
)  /\  ( B  =/=  P  /\  C  =/= 
P ) )  ->  A  =/=  P )
2 simplr 734 . . . . 5  |-  ( ( ( A  =/=  P  /\  B  =/=  P
)  /\  ( B  =/=  P  /\  C  =/= 
P ) )  ->  B  =/=  P )
3 simprr 736 . . . . 5  |-  ( ( ( A  =/=  P  /\  B  =/=  P
)  /\  ( B  =/=  P  /\  C  =/= 
P ) )  ->  C  =/=  P )
41, 2, 33jca 1137 . . . 4  |-  ( ( ( A  =/=  P  /\  B  =/=  P
)  /\  ( B  =/=  P  /\  C  =/= 
P ) )  -> 
( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
) )
5 simplr1 1002 . . . . . 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 1004 . . . . . 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 941 . . . . . . . . . . . 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 960 . . . . . . . . . . . . . 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 989 . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . 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 987 . . . . . . . . . . . . . 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 988 . . . . . . . . . . . . . 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 967 . . . . . . . . . . . . . 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 968 . . . . . . . . . . . . . 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 23989 . . . . . . . . . . . . 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 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
)  /\  A  Btwn  <. P ,  B >.  /\  B  Btwn  <. P ,  C >. ) )  -> 
( A  Btwn  <. P ,  C >.  \/  C  Btwn  <. P ,  A >. ) )
177, 16sylan2br 464 . . . . . . . . . . 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 601 . . . . . . . . . 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 742 . . . . . . . . . . . 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 736 . . . . . . . . . . . 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 24066 . . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 663 . . . . . . . . . . 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 601 . . . . . . . . . 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 371 . . . . . . . . 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 601 . . . . . . . 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 1000 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  B  Btwn  <. P ,  A >. )  /\  B  Btwn  <. P ,  C >. )  ->  B  =/=  P )
2928adantl 454 . . . . . . . . . . . . 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 2502 . . . . . . . . . . . 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 742 . . . . . . . . . . . 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 736 . . . . . . . . . . . 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 24064 . . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 1285 . . . . . . . . . . 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 601 . . . . . . . . . 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 941 . . . . . . . . . . . 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 968 . . . . . . . . . . . . . 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 967 . . . . . . . . . . . . . 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 23989 . . . . . . . . . . . . 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 384 . . . . . . . . . . . 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 464 . . . . . . . . . . 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 601 . . . . . . . . . 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 371 . . . . . . . . 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 601 . . . . . . . 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 371 . . . . . . 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 424 . . . . . 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 1137 . . . . 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 590 . . . 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 422 . 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 24085 . . . . 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 1189 . . . 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 24085 . . . . 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 1189 . . . 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 694 . . 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 941 . . . . 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 941 . . . . 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 681 . . . 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 800 . . . 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 245 . . 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 254 . 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 24085 . . 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 1189 . 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 261 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 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    e. wcel 1621    =/= wne 2419   <.cop 3584   class class class wbr 3963   ` cfv 4638   NNcn 9679   EEcee 23856    Btwn cbtwn 23857  OutsideOfcoutsideof 24082
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
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 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-oi 7158  df-card 7505  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-sum 12089  df-ee 23859  df-btwn 23860  df-cgr 23861  df-ofs 23946  df-ifs 24002  df-cgr3 24003  df-colinear 24004  df-fs 24005  df-outsideof 24083
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