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Theorem broutsideof3 29752
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 29748 . 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 457 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  N  e.  NN )
3 simpr3 1005 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
4 simpr1 1003 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  P  e.  ( EE `  N ) )
5 btwndiff 29653 . . . . . . . . . 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 1229 . . . . . . . . 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 465 . . . . . . . 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 976 . . . . . . . . . . . 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 978 . . . . . . . . . . . 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 1005 . . . . . . . . . . . . . 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 2714 . . . . . . . . . . . . 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 997 . . . . . . . . . . . . . 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 1032 . . . . . . . . . . . . . 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 1031 . . . . . . . . . . . . . 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 1030 . . . . . . . . . . . . . 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 999 . . . . . . . . . . . . . 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 1055 . . . . . . . . . . . . . . 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 29641 . . . . . . . . . . . . . 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 1004 . . . . . . . . . . . . . 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 29647 . . . . . . . . . . . . 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 1177 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 615 . . . . . . . . . 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 804 . . . . . . . . 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 2918 . . . . . . . 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 15 . . . . . . 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 615 . . . . . 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 1004 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
29 btwndiff 29653 . . . . . . . . . 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 1229 . . . . . . . . 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 465 . . . . . . . 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 978 . . . . . . . . . . . 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 1005 . . . . . . . . . . . . . 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 2714 . . . . . . . . . . . . 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 1004 . . . . . . . . . . . . 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 1055 . . . . . . . . . . . . . . 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 29641 . . . . . . . . . . . . . 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 29647 . . . . . . . . . . . . 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 1177 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 615 . . . . . . . . . 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 804 . . . . . . . . 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 2918 . . . . . . . 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 15 . . . . . . 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 615 . . . . . 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 380 . . . . 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 1045 . . . . . . . . 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 753 . . . . . . . . . 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 1039 . . . . . . . . . 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 1040 . . . . . . . . . 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 461 . . . . . . . . . 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 1046 . . . . . . . . . 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 29641 . . . . . . . . 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 1041 . . . . . . . . . 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 1047 . . . . . . . . . 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 29641 . . . . . . . . 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 29728 . . . . . . . . . . 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 1238 . . . . . . . . . 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 465 . . . . . . . . 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 1328 . . . . . . . 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 615 . . . . . . 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 804 . . . . . 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 2935 . . . . 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 191 . . . 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 641 . . 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 976 . . 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 976 . . 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 288 . 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 253 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 974    e. wcel 1804    =/= wne 2638   E.wrex 2794   <.cop 4020   class class class wbr 4437   ` cfv 5578   NNcn 10543   EEcee 24169    Btwn cbtwn 24170  OutsideOfcoutsideof 29745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-rp 11232  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-clim 13293  df-sum 13491  df-ee 24172  df-btwn 24173  df-cgr 24174  df-ofs 29609  df-colinear 29665  df-ifs 29666  df-cgr3 29667  df-fs 29668  df-outsideof 29746
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator