Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  broutsideof3 Unicode version

Theorem broutsideof3 25972
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 25968 . 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 444 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  N  e.  NN )
3 simpr3 965 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
4 simpr1 963 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  P  e.  ( EE `  N ) )
5 btwndiff 25873 . . . . . . . . . 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 1184 . . . . . . . . 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 452 . . . . . . . 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 938 . . . . . . . . . . . 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 940 . . . . . . . . . . . 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 965 . . . . . . . . . . . . . 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 2658 . . . . . . . . . . . . 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 957 . . . . . . . . . . . . . 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 992 . . . . . . . . . . . . . 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 991 . . . . . . . . . . . . . 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 990 . . . . . . . . . . . . . 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 959 . . . . . . . . . . . . . 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 1015 . . . . . . . . . . . . . . 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 25861 . . . . . . . . . . . . . 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 964 . . . . . . . . . . . . . 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 25867 . . . . . . . . . . . . 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 1134 . . . . . . . . . . . 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 467 . . . . . . . . . . 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 599 . . . . . . . . . 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 780 . . . . . . . . 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 2786 . . . . . . . 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 599 . . . . . 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 964 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
29 btwndiff 25873 . . . . . . . . . 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 1184 . . . . . . . . 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 452 . . . . . . . 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 940 . . . . . . . . . . . 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 965 . . . . . . . . . . . . . 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 2658 . . . . . . . . . . . . 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 964 . . . . . . . . . . . . 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 1015 . . . . . . . . . . . . . . 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 25861 . . . . . . . . . . . . . 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 25867 . . . . . . . . . . . . 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 1134 . . . . . . . . . . . 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 467 . . . . . . . . . . 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 599 . . . . . . . . . 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 780 . . . . . . . . 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 2786 . . . . . . . 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 599 . . . . . 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 370 . . . . 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 1005 . . . . . . . . 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 731 . . . . . . . . . 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 999 . . . . . . . . . 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 1000 . . . . . . . . . 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 448 . . . . . . . . . 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 1006 . . . . . . . . . 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 25861 . . . . . . . . 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 1001 . . . . . . . . . 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 1007 . . . . . . . . . 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 25861 . . . . . . . . 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 25948 . . . . . . . . . . 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 1193 . . . . . . . . . 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 452 . . . . . . . . 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 1282 . . . . . . . 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 599 . . . . . . 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 780 . . . . . 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 2798 . . . . 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 184 . . . 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 623 . . 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 938 . . 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 938 . . 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 280 . 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 245 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    e. wcel 1721    =/= wne 2575   E.wrex 2675   <.cop 3785   class class class wbr 4180   ` cfv 5421   NNcn 9964   EEcee 25739    Btwn cbtwn 25740  OutsideOfcoutsideof 25965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-sum 12443  df-ee 25742  df-btwn 25743  df-cgr 25744  df-ofs 25829  df-ifs 25885  df-cgr3 25886  df-colinear 25887  df-fs 25888  df-outsideof 25966
  Copyright terms: Public domain W3C validator