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Theorem outsideoftr 25777
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 25674 . . . . . . . . . . . . 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 25751 . . . . . . . . . . . . . 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 2633 . . . . . . . . . . . 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 25749 . . . . . . . . . . . . . 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 25674 . . . . . . . . . . . . 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 25770 . . . . 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 25770 . . . . 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 25770 . . 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 1717    =/= wne 2550   <.cop 3760   class class class wbr 4153   ` cfv 5394   NNcn 9932   EEcee 25541    Btwn cbtwn 25542  OutsideOfcoutsideof 25767
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407  df-ee 25544  df-btwn 25545  df-cgr 25546  df-ofs 25631  df-ifs 25687  df-cgr3 25688  df-colinear 25689  df-fs 25690  df-outsideof 25768
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