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Theorem broutsideof2 24155
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 24154 . 2  |-  ( POutsideOf <. A ,  B >.  <->  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )
2 btwntriv1 24049 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  A  Btwn  <. A ,  B >. )
323adant3r1 1160 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  Btwn  <. A ,  B >. )
4 breq1 4027 . . . . . . . 8  |-  ( A  =  P  ->  ( A  Btwn  <. A ,  B >.  <-> 
P  Btwn  <. A ,  B >. ) )
53, 4syl5ibcom 211 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( A  =  P  ->  P  Btwn  <. A ,  B >. ) )
65necon3bd 2484 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( -.  P  Btwn  <. A ,  B >.  ->  A  =/=  P ) )
76imp 418 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  -.  P  Btwn  <. A ,  B >. )  ->  A  =/=  P )
87adantrl 696 . . . 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 24045 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  B  Btwn  <. A ,  B >. )
1093adant3r1 1160 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  Btwn  <. A ,  B >. )
11 breq1 4027 . . . . . . . 8  |-  ( B  =  P  ->  ( B  Btwn  <. A ,  B >.  <-> 
P  Btwn  <. A ,  B >. ) )
1210, 11syl5ibcom 211 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( B  =  P  ->  P  Btwn  <. A ,  B >. ) )
1312necon3bd 2484 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( -.  P  Btwn  <. A ,  B >.  ->  B  =/=  P ) )
1413imp 418 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  -.  P  Btwn  <. A ,  B >. )  ->  B  =/=  P )
1514adantrl 696 . . . 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 24092 . . . . . 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 pm2.24 101 . . . . . . . 8  |-  ( P 
Btwn  <. A ,  B >.  ->  ( -.  P  Btwn  <. A ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
1817a1i 10 . . . . . . 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 939 . . . . . . . . . 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 24047 . . . . . . . . . 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 461 . . . . . . . . 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 374 . . . . . . . . 9  |-  ( A 
Btwn  <. P ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
2321, 22syl6bi 219 . . . . . . . 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 42 . . . . . . 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 373 . . . . . . . . 9  |-  ( B 
Btwn  <. P ,  A >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
2625a1d 22 . . . . . . . 8  |-  ( B 
Btwn  <. P ,  A >.  ->  ( -.  P  Btwn  <. A ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
2726a1i 10 . . . . . . 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 1246 . . . . . 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 206 . . . . 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 422 . . . 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 1132 . . 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 957 . . . . . 6  |-  ( ( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  -> 
( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
33 3ancomb 943 . . . . . . . 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 24103 . . . . . . . 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 461 . . . . . . 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 24102 . . . . . . 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 369 . . . . . 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 28 . . . . 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 418 . . . 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 962 . . . . . . . . . . 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 2463 . . . . . . . . . 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 1000 . . . . . . . . . . . 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 733 . . . . . . . . . . . 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 443 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  N  e.  NN )
45 simpr2 962 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
46 simpr1 961 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  P  e.  ( EE `  N ) )
47 simpr3 963 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
48 btwnswapid 24050 . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 660 . . . . . . . . . . 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 598 . . . . . . . . . 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 168 . . . . . . . . 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 1169 . . . . . . . 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 963 . . . . . . . . . . 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 2463 . . . . . . . . . 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 1000 . . . . . . . . . . . 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 733 . . . . . . . . . . . . 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 24048 . . . . . . . . . . . 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 939 . . . . . . . . . . . . . 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 24050 . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 660 . . . . . . . . . . 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 598 . . . . . . . . . 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 168 . . . . . . . . 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 1169 . . . . . . . 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 369 . . . . . . 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 27 . . . . . 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 78 . . . . 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 1166 . . . 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 518 . . 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 805 . 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 248 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 176    \/ wo 357    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1685    =/= wne 2447   <.cop 3644   class class class wbr 4024   ` cfv 5221   NNcn 9742   EEcee 23926    Btwn cbtwn 23927    Colinear ccolin 24070  OutsideOfcoutsideof 24152
This theorem is referenced by:  outsidene1  24156  outsidene2  24157  btwnoutside  24158  broutsideof3  24159  outsideofcom  24161  outsideoftr  24162  outsideofeq  24163  outsideofeu  24164  lineunray  24180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-oi 7221  df-card 7568  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-seq 11043  df-exp 11101  df-hash 11334  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-clim 11958  df-sum 12155  df-ee 23929  df-btwn 23930  df-cgr 23931  df-colinear 24074  df-outsideof 24153
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