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Theorem broutsideof3 24125
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 24121 . 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 445 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  N  e.  NN )
3 simpr3 968 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
4 simpr1 966 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  P  e.  ( EE `  N ) )
5 btwndiff 24026 . . . . . . . . . 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 1187 . . . . . . . . 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 453 . . . . . . . 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 941 . . . . . . . . . . . 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 943 . . . . . . . . . . . 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 968 . . . . . . . . . . . . . 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 2504 . . . . . . . . . . . . 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 960 . . . . . . . . . . . . . 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 995 . . . . . . . . . . . . . 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 994 . . . . . . . . . . . . . 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 993 . . . . . . . . . . . . . 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 962 . . . . . . . . . . . . . 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 1018 . . . . . . . . . . . . . . 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 24014 . . . . . . . . . . . . . 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 967 . . . . . . . . . . . . . 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 24020 . . . . . . . . . . . . 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 1137 . . . . . . . . . . . 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 468 . . . . . . . . . . 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 601 . . . . . . . . . 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 782 . . . . . . . . 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 2630 . . . . . . . 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 16 . . . . . . 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 601 . . . . . 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 967 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
29 btwndiff 24026 . . . . . . . . . 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 1187 . . . . . . . . 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 453 . . . . . . . 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 943 . . . . . . . . . . . 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 968 . . . . . . . . . . . . . 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 2504 . . . . . . . . . . . . 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 967 . . . . . . . . . . . . 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 1018 . . . . . . . . . . . . . . 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 24014 . . . . . . . . . . . . . 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 24020 . . . . . . . . . . . . 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 1137 . . . . . . . . . . . 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 468 . . . . . . . . . . 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 601 . . . . . . . . . 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 782 . . . . . . . . 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 2630 . . . . . . . 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 16 . . . . . . 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 601 . . . . . 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 371 . . . . 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 1008 . . . . . . . . 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 733 . . . . . . . . . 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 1002 . . . . . . . . . 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 1003 . . . . . . . . . 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 449 . . . . . . . . . 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 1009 . . . . . . . . . 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 24014 . . . . . . . . 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 1004 . . . . . . . . . 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 1010 . . . . . . . . . 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 24014 . . . . . . . . 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 24101 . . . . . . . . . . 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 1196 . . . . . . . . . 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 453 . . . . . . . . 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 1285 . . . . . . . 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 601 . . . . . . 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 782 . . . . . 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 2642 . . . . 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 185 . . . 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 625 . . 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 941 . . 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 941 . . 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 281 . 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 246 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 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    e. wcel 1621    =/= wne 2421   E.wrex 2519   <.cop 3617   class class class wbr 3997   ` cfv 4673   NNcn 9714   EEcee 23892    Btwn cbtwn 23893  OutsideOfcoutsideof 24118
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9934  df-z 9993  df-uz 10199  df-rp 10323  df-ico 10629  df-icc 10630  df-fz 10750  df-fzo 10838  df-seq 11014  df-exp 11072  df-hash 11305  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-clim 11928  df-sum 12125  df-ee 23895  df-btwn 23896  df-cgr 23897  df-ofs 23982  df-ifs 24038  df-cgr3 24039  df-colinear 24040  df-fs 24041  df-outsideof 24119
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