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Theorem broutsideof3 28104
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 28100 . 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 996 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
4 simpr1 994 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  P  e.  ( EE `  N ) )
5 btwndiff 28005 . . . . . . . . . 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 1218 . . . . . . . . 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 967 . . . . . . . . . . . 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 969 . . . . . . . . . . . 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 996 . . . . . . . . . . . . . 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 2689 . . . . . . . . . . . . 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 988 . . . . . . . . . . . . . 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 1023 . . . . . . . . . . . . . 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 1022 . . . . . . . . . . . . . 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 1021 . . . . . . . . . . . . . 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 990 . . . . . . . . . . . . . 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 1046 . . . . . . . . . . . . . . 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 27993 . . . . . . . . . . . . . 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 995 . . . . . . . . . . . . . 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 27999 . . . . . . . . . . . . 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 1168 . . . . . . . . . . . 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 802 . . . . . . . . 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 2822 . . . . . . . 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 995 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
29 btwndiff 28005 . . . . . . . . . 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 1218 . . . . . . . . 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 969 . . . . . . . . . . . 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 996 . . . . . . . . . . . . . 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 2689 . . . . . . . . . . . . 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 995 . . . . . . . . . . . . 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 1046 . . . . . . . . . . . . . . 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 27993 . . . . . . . . . . . . . 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 27999 . . . . . . . . . . . . 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 1168 . . . . . . . . . . . 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 802 . . . . . . . . 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 2822 . . . . . . . 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 1036 . . . . . . . . 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 1030 . . . . . . . . . 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 1031 . . . . . . . . . 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 1037 . . . . . . . . . 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 27993 . . . . . . . . 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 1032 . . . . . . . . . 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 1038 . . . . . . . . . 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 27993 . . . . . . . . 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 28080 . . . . . . . . . . 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 1227 . . . . . . . . . 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 1317 . . . . . . . 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 802 . . . . . 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 2835 . . . . 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 967 . . 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 967 . . 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 965    e. wcel 1756    =/= wne 2600   E.wrex 2710   <.cop 3876   class class class wbr 4285   ` cfv 5411   NNcn 10314   EEcee 23079    Btwn cbtwn 23080  OutsideOfcoutsideof 28097
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 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-se 4672  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-isom 5420  df-riota 6045  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 23082  df-btwn 23083  df-cgr 23084  df-ofs 27961  df-colinear 28017  df-ifs 28018  df-cgr3 28019  df-fs 28020  df-outsideof 28098
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator