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Theorem broutsideof2 25763
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 25762 . 2  |-  ( POutsideOf <. A ,  B >.  <->  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )
2 btwntriv1 25657 . . . . . . . . 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 4149 . . . . . . . 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 2580 . . . . . 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 25653 . . . . . . . . 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 4149 . . . . . . . 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 2580 . . . . . 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 25700 . . . . . 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 25655 . . . . . . . . . 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 25711 . . . . . . . 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 25710 . . . . . . 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 2559 . . . . . . . . . 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 25658 . . . . . . . . . . . . . 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 2559 . . . . . . . . . 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 25656 . . . . . . . . . . . 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 25658 . . . . . . . . . . . . . 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 1717    =/= wne 2543   <.cop 3753   class class class wbr 4146   ` cfv 5387   NNcn 9925   EEcee 25534    Btwn cbtwn 25535    Colinear ccolin 25678  OutsideOfcoutsideof 25760
This theorem is referenced by:  outsidene1  25764  outsidene2  25765  btwnoutside  25766  broutsideof3  25767  outsideofcom  25769  outsideoftr  25770  outsideofeq  25771  outsideofeu  25772  lineunray  25788
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-ico 10847  df-icc 10848  df-fz 10969  df-fzo 11059  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-sum 12400  df-ee 25537  df-btwn 25538  df-cgr 25539  df-colinear 25682  df-outsideof 25761
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