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Theorem broutsideof3 24159
Description: Characterization of outsideness in terms of relationship to a fourth point. Theorem 6.3 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
broutsideof3  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. A ,  B >.  <-> 
( A  =/=  P  /\  B  =/=  P  /\  E. c  e.  ( EE `  N ) ( c  =/=  P  /\  P  Btwn  <. A , 
c >.  /\  P  Btwn  <. B ,  c >. ) ) ) )
Distinct variable groups:    N, c    A, c    B, c    P, c

Proof of Theorem broutsideof3
StepHypRef Expression
1 broutsideof2 24155 . 2  |-  ( ( 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 >. ) ) ) )
2 simpl 443 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  N  e.  NN )
3 simpr3 963 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
4 simpr1 961 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  P  e.  ( EE `  N ) )
5 btwndiff 24060 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  B  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) )  ->  E. c  e.  ( EE `  N
) ( P  Btwn  <. B ,  c >.  /\  P  =/=  c ) )
62, 3, 4, 5syl3anc 1182 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  E. c  e.  ( EE `  N ) ( P  Btwn  <. B , 
c >.  /\  P  =/=  c ) )
76adantr 451 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  A  Btwn  <. P ,  B >. ) )  ->  E. c  e.  ( EE `  N
) ( P  Btwn  <. B ,  c >.  /\  P  =/=  c ) )
8 df-3an 936 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  c  e.  ( EE `  N
) )  <->  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) ) )
9 3anass 938 . . . . . . . . . . . 12  |-  ( ( ( ( A  =/= 
P  /\  B  =/=  P )  /\  A  Btwn  <. P ,  B >. )  /\  P  Btwn  <. B , 
c >.  /\  P  =/=  c )  <->  ( (
( A  =/=  P  /\  B  =/=  P
)  /\  A  Btwn  <. P ,  B >. )  /\  ( P  Btwn  <. B ,  c >.  /\  P  =/=  c ) ) )
10 simpr3 963 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  A  Btwn  <. P ,  B >. )  /\  P  Btwn  <. B , 
c >.  /\  P  =/=  c ) )  ->  P  =/=  c )
1110necomd 2530 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  A  Btwn  <. P ,  B >. )  /\  P  Btwn  <. B , 
c >.  /\  P  =/=  c ) )  -> 
c  =/=  P )
12 simp1 955 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  c  e.  ( EE `  N
) )  ->  N  e.  NN )
13 simp23 990 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  c  e.  ( EE `  N
) )  ->  B  e.  ( EE `  N
) )
14 simp22 989 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  c  e.  ( EE `  N
) )  ->  A  e.  ( EE `  N
) )
15 simp21 988 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  c  e.  ( EE `  N
) )  ->  P  e.  ( EE `  N
) )
16 simp3 957 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  c  e.  ( EE `  N
) )  ->  c  e.  ( EE `  N
) )
17 simpr1r 1013 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  A  Btwn  <. P ,  B >. )  /\  P  Btwn  <. B , 
c >.  /\  P  =/=  c ) )  ->  A  Btwn  <. P ,  B >. )
1812, 14, 15, 13, 17btwncomand 24048 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  A  Btwn  <. P ,  B >. )  /\  P  Btwn  <. B , 
c >.  /\  P  =/=  c ) )  ->  A  Btwn  <. B ,  P >. )
19 simpr2 962 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  A  Btwn  <. P ,  B >. )  /\  P  Btwn  <. B , 
c >.  /\  P  =/=  c ) )  ->  P  Btwn  <. B ,  c
>. )
2012, 13, 14, 15, 16, 18, 19btwnexch3and 24054 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  A  Btwn  <. P ,  B >. )  /\  P  Btwn  <. B , 
c >.  /\  P  =/=  c ) )  ->  P  Btwn  <. A ,  c
>. )
2111, 20, 193jca 1132 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  A  Btwn  <. P ,  B >. )  /\  P  Btwn  <. B , 
c >.  /\  P  =/=  c ) )  -> 
( c  =/=  P  /\  P  Btwn  <. A , 
c >.  /\  P  Btwn  <. B ,  c >. ) )
228, 9, 21syl2anbr 466 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  A  Btwn  <. P ,  B >. )  /\  ( P  Btwn  <. B ,  c >.  /\  P  =/=  c ) ) )  ->  (
c  =/=  P  /\  P  Btwn  <. A ,  c
>.  /\  P  Btwn  <. B , 
c >. ) )
2322expr 598 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  A  Btwn  <. P ,  B >. ) )  ->  ( ( P  Btwn  <. B ,  c
>.  /\  P  =/=  c
)  ->  ( c  =/=  P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )
2423an32s 779 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  A  Btwn  <. P ,  B >. ) )  /\  c  e.  ( EE `  N
) )  ->  (
( P  Btwn  <. B , 
c >.  /\  P  =/=  c )  ->  (
c  =/=  P  /\  P  Btwn  <. A ,  c
>.  /\  P  Btwn  <. B , 
c >. ) ) )
2524reximdva 2656 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  A  Btwn  <. P ,  B >. ) )  ->  ( E. c  e.  ( EE `  N ) ( P 
Btwn  <. B ,  c
>.  /\  P  =/=  c
)  ->  E. c  e.  ( EE `  N
) ( c  =/= 
P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )
267, 25mpd 14 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  A  Btwn  <. P ,  B >. ) )  ->  E. c  e.  ( EE `  N
) ( c  =/= 
P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) )
2726expr 598 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( A  =/=  P  /\  B  =/=  P ) )  -> 
( A  Btwn  <. P ,  B >.  ->  E. c  e.  ( EE `  N
) ( c  =/= 
P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )
28 simpr2 962 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
29 btwndiff 24060 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) )  ->  E. c  e.  ( EE `  N
) ( P  Btwn  <. A ,  c >.  /\  P  =/=  c ) )
302, 28, 4, 29syl3anc 1182 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  E. c  e.  ( EE `  N ) ( P  Btwn  <. A , 
c >.  /\  P  =/=  c ) )
3130adantr 451 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  B  Btwn  <. P ,  A >. ) )  ->  E. c  e.  ( EE `  N
) ( P  Btwn  <. A ,  c >.  /\  P  =/=  c ) )
32 3anass 938 . . . . . . . . . . . 12  |-  ( ( ( ( A  =/= 
P  /\  B  =/=  P )  /\  B  Btwn  <. P ,  A >. )  /\  P  Btwn  <. A , 
c >.  /\  P  =/=  c )  <->  ( (
( A  =/=  P  /\  B  =/=  P
)  /\  B  Btwn  <. P ,  A >. )  /\  ( P  Btwn  <. A ,  c >.  /\  P  =/=  c ) ) )
33 simpr3 963 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  B  Btwn  <. P ,  A >. )  /\  P  Btwn  <. A , 
c >.  /\  P  =/=  c ) )  ->  P  =/=  c )
3433necomd 2530 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  B  Btwn  <. P ,  A >. )  /\  P  Btwn  <. A , 
c >.  /\  P  =/=  c ) )  -> 
c  =/=  P )
35 simpr2 962 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  B  Btwn  <. P ,  A >. )  /\  P  Btwn  <. A , 
c >.  /\  P  =/=  c ) )  ->  P  Btwn  <. A ,  c
>. )
36 simpr1r 1013 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  B  Btwn  <. P ,  A >. )  /\  P  Btwn  <. A , 
c >.  /\  P  =/=  c ) )  ->  B  Btwn  <. P ,  A >. )
3712, 13, 15, 14, 36btwncomand 24048 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  B  Btwn  <. P ,  A >. )  /\  P  Btwn  <. A , 
c >.  /\  P  =/=  c ) )  ->  B  Btwn  <. A ,  P >. )
3812, 14, 13, 15, 16, 37, 35btwnexch3and 24054 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  B  Btwn  <. P ,  A >. )  /\  P  Btwn  <. A , 
c >.  /\  P  =/=  c ) )  ->  P  Btwn  <. B ,  c
>. )
3934, 35, 383jca 1132 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  B  Btwn  <. P ,  A >. )  /\  P  Btwn  <. A , 
c >.  /\  P  =/=  c ) )  -> 
( c  =/=  P  /\  P  Btwn  <. A , 
c >.  /\  P  Btwn  <. B ,  c >. ) )
408, 32, 39syl2anbr 466 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  /\  (
( ( A  =/= 
P  /\  B  =/=  P )  /\  B  Btwn  <. P ,  A >. )  /\  ( P  Btwn  <. A ,  c >.  /\  P  =/=  c ) ) )  ->  (
c  =/=  P  /\  P  Btwn  <. A ,  c
>.  /\  P  Btwn  <. B , 
c >. ) )
4140expr 598 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  B  Btwn  <. P ,  A >. ) )  ->  ( ( P  Btwn  <. A ,  c
>.  /\  P  =/=  c
)  ->  ( c  =/=  P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )
4241an32s 779 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  B  Btwn  <. P ,  A >. ) )  /\  c  e.  ( EE `  N
) )  ->  (
( P  Btwn  <. A , 
c >.  /\  P  =/=  c )  ->  (
c  =/=  P  /\  P  Btwn  <. A ,  c
>.  /\  P  Btwn  <. B , 
c >. ) ) )
4342reximdva 2656 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  B  Btwn  <. P ,  A >. ) )  ->  ( E. c  e.  ( EE `  N ) ( P 
Btwn  <. A ,  c
>.  /\  P  =/=  c
)  ->  E. c  e.  ( EE `  N
) ( c  =/= 
P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )
4431, 43mpd 14 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  B  Btwn  <. P ,  A >. ) )  ->  E. c  e.  ( EE `  N
) ( c  =/= 
P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) )
4544expr 598 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( A  =/=  P  /\  B  =/=  P ) )  -> 
( B  Btwn  <. P ,  A >.  ->  E. c  e.  ( EE `  N
) ( c  =/= 
P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )
4627, 45jaod 369 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( A  =/=  P  /\  B  =/=  P ) )  -> 
( ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  ->  E. c  e.  ( EE `  N
) ( c  =/= 
P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )
47 simprr1 1003 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  ( c  =/=  P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )  ->  c  =/=  P
)
48 simpll 730 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  ->  N  e.  NN )
49 simplr1 997 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  ->  P  e.  ( EE `  N
) )
50 simplr2 998 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  ->  A  e.  ( EE `  N
) )
51 simpr 447 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  ->  c  e.  ( EE `  N
) )
52 simprr2 1004 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  ( c  =/=  P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )  ->  P  Btwn  <. A , 
c >. )
5348, 49, 50, 51, 52btwncomand 24048 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  ( c  =/=  P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )  ->  P  Btwn  <. c ,  A >. )
54 simplr3 999 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  ->  B  e.  ( EE `  N
) )
55 simprr3 1005 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  ( c  =/=  P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )  ->  P  Btwn  <. B , 
c >. )
5648, 49, 54, 51, 55btwncomand 24048 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  ( c  =/=  P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )  ->  P  Btwn  <. c ,  B >. )
57 btwnconn2 24135 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( c  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( ( c  =/= 
P  /\  P  Btwn  <.
c ,  A >.  /\  P  Btwn  <. c ,  B >. )  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
5848, 51, 49, 50, 54, 57syl122anc 1191 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  ->  (
( c  =/=  P  /\  P  Btwn  <. c ,  A >.  /\  P  Btwn  <.
c ,  B >. )  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
5958adantr 451 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  ( c  =/=  P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )  ->  ( ( c  =/=  P  /\  P  Btwn  <. c ,  A >.  /\  P  Btwn  <. c ,  B >. )  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
6047, 53, 56, 59mp3and 1280 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  /\  (
( A  =/=  P  /\  B  =/=  P
)  /\  ( c  =/=  P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
6160expr 598 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  c  e.  ( EE `  N
) )  /\  ( A  =/=  P  /\  B  =/=  P ) )  -> 
( ( c  =/= 
P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. )  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
6261an32s 779 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  /\  ( A  =/=  P  /\  B  =/=  P ) )  /\  c  e.  ( EE `  N ) )  -> 
( ( c  =/= 
P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. )  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
6362rexlimdva 2668 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( A  =/=  P  /\  B  =/=  P ) )  -> 
( E. c  e.  ( EE `  N
) ( c  =/= 
P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. )  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
6446, 63impbid 183 . . . 4  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( A  =/=  P  /\  B  =/=  P ) )  -> 
( ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  <->  E. c  e.  ( EE `  N
) ( c  =/= 
P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) )
6564pm5.32da 622 . . 3  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( ( ( A  =/=  P  /\  B  =/=  P )  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  <->  ( ( A  =/=  P  /\  B  =/=  P )  /\  E. c  e.  ( EE `  N ) ( c  =/=  P  /\  P  Btwn  <. A ,  c
>.  /\  P  Btwn  <. B , 
c >. ) ) ) )
66 df-3an 936 . . 3  |-  ( ( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  <->  ( ( A  =/=  P  /\  B  =/=  P )  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
67 df-3an 936 . . 3  |-  ( ( A  =/=  P  /\  B  =/=  P  /\  E. c  e.  ( EE `  N ) ( c  =/=  P  /\  P  Btwn  <. A ,  c
>.  /\  P  Btwn  <. B , 
c >. ) )  <->  ( ( A  =/=  P  /\  B  =/=  P )  /\  E. c  e.  ( EE `  N ) ( c  =/=  P  /\  P  Btwn  <. A ,  c
>.  /\  P  Btwn  <. B , 
c >. ) ) )
6865, 66, 673bitr4g 279 . 2  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( ( A  =/= 
P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  <->  ( A  =/=  P  /\  B  =/= 
P  /\  E. c  e.  ( EE `  N
) ( c  =/= 
P  /\  P  Btwn  <. A ,  c >.  /\  P  Btwn  <. B , 
c >. ) ) ) )
691, 68bitrd 244 1  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. A ,  B >.  <-> 
( A  =/=  P  /\  B  =/=  P  /\  E. c  e.  ( EE `  N ) ( c  =/=  P  /\  P  Btwn  <. A , 
c >.  /\  P  Btwn  <. B ,  c >. ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    e. wcel 1685    =/= wne 2447   E.wrex 2545   <.cop 3644   class class class wbr 4024   ` cfv 5221   NNcn 9742   EEcee 23926    Btwn cbtwn 23927  OutsideOfcoutsideof 24152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-oi 7221  df-card 7568  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-seq 11043  df-exp 11101  df-hash 11334  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-clim 11958  df-sum 12155  df-ee 23929  df-btwn 23930  df-cgr 23931  df-ofs 24016  df-ifs 24072  df-cgr3 24073  df-colinear 24074  df-fs 24075  df-outsideof 24153
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