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Theorem broutsideof2 25960
Description: Alternate form of OutsideOf. Definition 6.1 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
broutsideof2  |-  ( ( 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 >. ) ) ) )

Proof of Theorem broutsideof2
StepHypRef Expression
1 broutsideof 25959 . 2  |-  ( POutsideOf <. A ,  B >.  <->  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )
2 btwntriv1 25854 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  A  Btwn  <. A ,  B >. )
323adant3r1 1162 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  Btwn  <. A ,  B >. )
4 breq1 4175 . . . . . . . 8  |-  ( A  =  P  ->  ( A  Btwn  <. A ,  B >.  <-> 
P  Btwn  <. A ,  B >. ) )
53, 4syl5ibcom 212 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( A  =  P  ->  P  Btwn  <. A ,  B >. ) )
65necon3bd 2604 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( -.  P  Btwn  <. A ,  B >.  ->  A  =/=  P ) )
76imp 419 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  -.  P  Btwn  <. A ,  B >. )  ->  A  =/=  P )
87adantrl 697 . . . 4  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )  ->  A  =/=  P )
9 btwntriv2 25850 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  B  Btwn  <. A ,  B >. )
1093adant3r1 1162 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  Btwn  <. A ,  B >. )
11 breq1 4175 . . . . . . . 8  |-  ( B  =  P  ->  ( B  Btwn  <. A ,  B >.  <-> 
P  Btwn  <. A ,  B >. ) )
1210, 11syl5ibcom 212 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( B  =  P  ->  P  Btwn  <. A ,  B >. ) )
1312necon3bd 2604 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( -.  P  Btwn  <. A ,  B >.  ->  B  =/=  P ) )
1413imp 419 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  -.  P  Btwn  <. A ,  B >. )  ->  B  =/=  P )
1514adantrl 697 . . . 4  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )  ->  B  =/=  P )
16 brcolinear 25897 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( P  Colinear  <. A ,  B >. 
<->  ( P  Btwn  <. A ,  B >.  \/  A  Btwn  <. B ,  P >.  \/  B  Btwn  <. P ,  A >. ) ) )
17 pm2.24 103 . . . . . . . 8  |-  ( P 
Btwn  <. A ,  B >.  ->  ( -.  P  Btwn  <. A ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
1817a1i 11 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( P  Btwn  <. A ,  B >.  ->  ( -.  P  Btwn  <. A ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
19 3anrot 941 . . . . . . . . . 10  |-  ( ( P  e.  ( EE
`  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  <->  ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )
20 btwncom 25852 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. B ,  P >. 
<->  A  Btwn  <. P ,  B >. ) )
2119, 20sylan2b 462 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. B ,  P >. 
<->  A  Btwn  <. P ,  B >. ) )
22 orc 375 . . . . . . . . 9  |-  ( A 
Btwn  <. P ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
2321, 22syl6bi 220 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. B ,  P >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
2423a1dd 44 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. B ,  P >.  ->  ( -.  P  Btwn  <. A ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
25 olc 374 . . . . . . . . 9  |-  ( B 
Btwn  <. P ,  A >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
2625a1d 23 . . . . . . . 8  |-  ( B 
Btwn  <. P ,  A >.  ->  ( -.  P  Btwn  <. A ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
2726a1i 11 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. P ,  A >.  ->  ( -.  P  Btwn  <. A ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
2818, 24, 273jaod 1248 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( ( P  Btwn  <. A ,  B >.  \/  A  Btwn  <. B ,  P >.  \/  B  Btwn  <. P ,  A >. )  ->  ( -.  P  Btwn  <. A ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
2916, 28sylbid 207 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( P  Colinear  <. A ,  B >.  ->  ( -.  P  Btwn  <. A ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
3029imp32 423 . . . 4  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
318, 15, 303jca 1134 . . 3  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )  ->  ( A  =/=  P  /\  B  =/= 
P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
32 simp3 959 . . . . . 6  |-  ( ( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  -> 
( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
33 3ancomb 945 . . . . . . . 8  |-  ( ( P  e.  ( EE
`  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  <->  ( P  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) ) )
34 btwncolinear2 25908 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. P ,  B >.  ->  P  Colinear  <. A ,  B >. ) )
3533, 34sylan2b 462 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. P ,  B >.  ->  P  Colinear  <. A ,  B >. ) )
36 btwncolinear1 25907 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. P ,  A >.  ->  P  Colinear  <. A ,  B >. ) )
3735, 36jaod 370 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  ->  P  Colinear  <. A ,  B >. ) )
3832, 37syl5 30 . . . . 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 >. ) )  ->  P  Colinear  <. A ,  B >. ) )
3938imp 419 . . . 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 >. ) ) )  ->  P  Colinear  <. A ,  B >. )
40 simpr2 964 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( A  Btwn  <. P ,  B >.  /\  A  =/=  P  /\  B  =/=  P
) )  ->  A  =/=  P )
4140neneqd 2583 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( A  Btwn  <. P ,  B >.  /\  A  =/=  P  /\  B  =/=  P
) )  ->  -.  A  =  P )
42 simprl1 1002 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( A  Btwn  <. P ,  B >.  /\  A  =/=  P  /\  B  =/=  P
)  /\  P  Btwn  <. A ,  B >. ) )  ->  A  Btwn  <. P ,  B >. )
43 simprr 734 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( A  Btwn  <. P ,  B >.  /\  A  =/=  P  /\  B  =/=  P
)  /\  P  Btwn  <. A ,  B >. ) )  ->  P  Btwn  <. A ,  B >. )
44 simpl 444 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  N  e.  NN )
45 simpr2 964 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
46 simpr1 963 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  P  e.  ( EE `  N ) )
47 simpr3 965 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
48 btwnswapid 25855 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  P  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <. P ,  B >.  /\  P  Btwn  <. A ,  B >. )  ->  A  =  P ) )
4944, 45, 46, 47, 48syl13anc 1186 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <. P ,  B >.  /\  P  Btwn  <. A ,  B >. )  ->  A  =  P ) )
5049adantr 452 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( A  Btwn  <. P ,  B >.  /\  A  =/=  P  /\  B  =/=  P
)  /\  P  Btwn  <. A ,  B >. ) )  ->  ( ( A  Btwn  <. P ,  B >.  /\  P  Btwn  <. A ,  B >. )  ->  A  =  P ) )
5142, 43, 50mp2and 661 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( A  Btwn  <. P ,  B >.  /\  A  =/=  P  /\  B  =/=  P
)  /\  P  Btwn  <. A ,  B >. ) )  ->  A  =  P )
5251expr 599 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( A  Btwn  <. P ,  B >.  /\  A  =/=  P  /\  B  =/=  P
) )  ->  ( P  Btwn  <. A ,  B >.  ->  A  =  P ) )
5341, 52mtod 170 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( A  Btwn  <. P ,  B >.  /\  A  =/=  P  /\  B  =/=  P
) )  ->  -.  P  Btwn  <. A ,  B >. )
54533exp2 1171 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. P ,  B >.  ->  ( A  =/=  P  ->  ( B  =/=  P  ->  -.  P  Btwn  <. A ,  B >. ) ) ) )
55 simpr3 965 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. P ,  A >.  /\  A  =/=  P  /\  B  =/=  P
) )  ->  B  =/=  P )
5655neneqd 2583 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. P ,  A >.  /\  A  =/=  P  /\  B  =/=  P
) )  ->  -.  B  =  P )
57 simprl1 1002 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. P ,  A >.  /\  A  =/=  P  /\  B  =/=  P
)  /\  P  Btwn  <. A ,  B >. ) )  ->  B  Btwn  <. P ,  A >. )
58 simprr 734 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. P ,  A >.  /\  A  =/=  P  /\  B  =/=  P
)  /\  P  Btwn  <. A ,  B >. ) )  ->  P  Btwn  <. A ,  B >. )
5944, 46, 45, 47, 58btwncomand 25853 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. P ,  A >.  /\  A  =/=  P  /\  B  =/=  P
)  /\  P  Btwn  <. A ,  B >. ) )  ->  P  Btwn  <. B ,  A >. )
60 3anrot 941 . . . . . . . . . . . . . 14  |-  ( ( B  e.  ( EE
`  N )  /\  P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  <->  ( P  e.  ( EE `  N
)  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )
61 btwnswapid 25855 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) ) )  -> 
( ( B  Btwn  <. P ,  A >.  /\  P  Btwn  <. B ,  A >. )  ->  B  =  P ) )
6260, 61sylan2br 463 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( ( B  Btwn  <. P ,  A >.  /\  P  Btwn  <. B ,  A >. )  ->  B  =  P ) )
6362adantr 452 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. P ,  A >.  /\  A  =/=  P  /\  B  =/=  P
)  /\  P  Btwn  <. A ,  B >. ) )  ->  ( ( B  Btwn  <. P ,  A >.  /\  P  Btwn  <. B ,  A >. )  ->  B  =  P ) )
6457, 59, 63mp2and 661 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. P ,  A >.  /\  A  =/=  P  /\  B  =/=  P
)  /\  P  Btwn  <. A ,  B >. ) )  ->  B  =  P )
6564expr 599 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. P ,  A >.  /\  A  =/=  P  /\  B  =/=  P
) )  ->  ( P  Btwn  <. A ,  B >.  ->  B  =  P ) )
6656, 65mtod 170 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  ( B  Btwn  <. P ,  A >.  /\  A  =/=  P  /\  B  =/=  P
) )  ->  -.  P  Btwn  <. A ,  B >. )
67663exp2 1171 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. P ,  A >.  ->  ( A  =/=  P  ->  ( B  =/=  P  ->  -.  P  Btwn  <. A ,  B >. ) ) ) )
6854, 67jaod 370 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  ->  ( A  =/=  P  ->  ( B  =/=  P  ->  -.  P  Btwn  <. A ,  B >. ) ) ) )
6968com12 29 . . . . . 6  |-  ( ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  ->  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  ->  ( A  =/=  P  ->  ( B  =/=  P  ->  -.  P  Btwn  <. A ,  B >. ) ) ) )
7069com4l 80 . . . . 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 >. )  ->  -.  P  Btwn  <. A ,  B >. ) ) ) )
71703imp2 1168 . . . 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 >. ) ) )  ->  -.  P  Btwn  <. A ,  B >. )
7239, 71jca 519 . . 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 >. ) ) )  ->  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )
7331, 72impbida 806 . 2  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. )  <->  ( A  =/= 
P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
741, 73syl5bb 249 1  |-  ( ( 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 >. ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   <.cop 3777   class class class wbr 4172   ` cfv 5413   NNcn 9956   EEcee 25731    Btwn cbtwn 25732    Colinear ccolin 25875  OutsideOfcoutsideof 25957
This theorem is referenced by:  outsidene1  25961  outsidene2  25962  btwnoutside  25963  broutsideof3  25964  outsideofcom  25966  outsideoftr  25967  outsideofeq  25968  outsideofeu  25969  lineunray  25985
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-ee 25734  df-btwn 25735  df-cgr 25736  df-colinear 25879  df-outsideof 25958
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