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Theorem outsideoftr 28129
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 753 . . . . 5  |-  ( ( ( A  =/=  P  /\  B  =/=  P
)  /\  ( B  =/=  P  /\  C  =/= 
P ) )  ->  A  =/=  P )
2 simplr 754 . . . . 5  |-  ( ( ( A  =/=  P  /\  B  =/=  P
)  /\  ( B  =/=  P  /\  C  =/= 
P ) )  ->  B  =/=  P )
3 simprr 756 . . . . 5  |-  ( ( ( A  =/=  P  /\  B  =/=  P
)  /\  ( B  =/=  P  /\  C  =/= 
P ) )  ->  C  =/=  P )
41, 2, 33jca 1168 . . . 4  |-  ( ( ( A  =/=  P  /\  B  =/=  P
)  /\  ( B  =/=  P  /\  C  =/= 
P ) )  -> 
( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
) )
5 simplr1 1030 . . . . . 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 1032 . . . . . 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 967 . . . . . . . . . . . 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 988 . . . . . . . . . . . . . 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 1017 . . . . . . . . . . . . . 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 1014 . . . . . . . . . . . . . 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 1015 . . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . . 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 995 . . . . . . . . . . . . . 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 996 . . . . . . . . . . . . . 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 28026 . . . . . . . . . . . . 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 392 . . . . . . . . . . . 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 476 . . . . . . . . . . 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 615 . . . . . . . . . 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 762 . . . . . . . . . . . 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 756 . . . . . . . . . . . 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 28103 . . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 679 . . . . . . . . . . 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 615 . . . . . . . . . 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 380 . . . . . . . . 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 615 . . . . . . . 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 1028 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  B  Btwn  <. P ,  A >. )  /\  B  Btwn  <. P ,  C >. )  ->  B  =/=  P )
2928adantl 466 . . . . . . . . . . . . 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 2690 . . . . . . . . . . . 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 762 . . . . . . . . . . . 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 756 . . . . . . . . . . . 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 28101 . . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 1317 . . . . . . . . . . 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 615 . . . . . . . . . 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 967 . . . . . . . . . . . 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 996 . . . . . . . . . . . . . 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 995 . . . . . . . . . . . . . 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 28026 . . . . . . . . . . . . 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 393 . . . . . . . . . . . 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 476 . . . . . . . . . . 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 615 . . . . . . . . . 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 380 . . . . . . . . 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 615 . . . . . . . 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 380 . . . . . . 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 433 . . . . . 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 1168 . . . . 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 604 . . . 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 32 . . 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 >. ) ) ) ) )
5251impd 431 . 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 28122 . . . . 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 1220 . . . 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 28122 . . . . 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 1220 . . . 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 710 . . 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 967 . . . . 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 967 . . . . 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 697 . . . 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 820 . . . 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 252 . . 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 261 . 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 28122 . . 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 1220 . 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 268 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    e. wcel 1756    =/= wne 2601   <.cop 3878   class class class wbr 4287   ` cfv 5413   NNcn 10314   EEcee 23102    Btwn cbtwn 23103  OutsideOfcoutsideof 28119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-ee 23105  df-btwn 23106  df-cgr 23107  df-ofs 27983  df-colinear 28039  df-ifs 28040  df-cgr3 28041  df-fs 28042  df-outsideof 28120
This theorem is referenced by: (None)
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