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Theorem broutsideof2 24085
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 24084 . 2  |-  ( POutsideOf <. A ,  B >.  <->  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )
2 btwntriv1 23979 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  A  Btwn  <. A ,  B >. )
323adant3r1 1165 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  Btwn  <. A ,  B >. )
4 breq1 3966 . . . . . . . 8  |-  ( A  =  P  ->  ( A  Btwn  <. A ,  B >.  <-> 
P  Btwn  <. A ,  B >. ) )
53, 4syl5ibcom 213 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( A  =  P  ->  P  Btwn  <. A ,  B >. ) )
65necon3bd 2456 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( -.  P  Btwn  <. A ,  B >.  ->  A  =/=  P ) )
76imp 420 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  -.  P  Btwn  <. A ,  B >. )  ->  A  =/=  P )
87adantrl 699 . . . 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 23975 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  B  Btwn  <. A ,  B >. )
1093adant3r1 1165 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  Btwn  <. A ,  B >. )
11 breq1 3966 . . . . . . . 8  |-  ( B  =  P  ->  ( B  Btwn  <. A ,  B >.  <-> 
P  Btwn  <. A ,  B >. ) )
1210, 11syl5ibcom 213 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( B  =  P  ->  P  Btwn  <. A ,  B >. ) )
1312necon3bd 2456 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( -.  P  Btwn  <. A ,  B >.  ->  B  =/=  P ) )
1413imp 420 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  -.  P  Btwn  <. A ,  B >. )  ->  B  =/=  P )
1514adantrl 699 . . . 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 24022 . . . . . 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 12 . . . . . . 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 944 . . . . . . . . . 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 23977 . . . . . . . . . 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 463 . . . . . . . . 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 376 . . . . . . . . 9  |-  ( A 
Btwn  <. P ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
2321, 22syl6bi 221 . . . . . . . 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 375 . . . . . . . . 9  |-  ( B 
Btwn  <. P ,  A >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
2625a1d 24 . . . . . . . 8  |-  ( B 
Btwn  <. P ,  A >.  ->  ( -.  P  Btwn  <. A ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
2726a1i 12 . . . . . . 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 1251 . . . . . 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 208 . . . . 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 424 . . . 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 1137 . . 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 962 . . . . . 6  |-  ( ( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  -> 
( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
33 3ancomb 948 . . . . . . . 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 24033 . . . . . . . 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 463 . . . . . . 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 24032 . . . . . . 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 371 . . . . . 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 420 . . . 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 967 . . . . . . . . . . 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 2435 . . . . . . . . . 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 1005 . . . . . . . . . . . 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 736 . . . . . . . . . . . 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 445 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  N  e.  NN )
45 simpr2 967 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
46 simpr1 966 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  P  e.  ( EE `  N ) )
47 simpr3 968 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
48 btwnswapid 23980 . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 663 . . . . . . . . . . 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 601 . . . . . . . . . 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 1174 . . . . . . . 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 968 . . . . . . . . . . 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 2435 . . . . . . . . . 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 1005 . . . . . . . . . . . 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 736 . . . . . . . . . . . . 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 23978 . . . . . . . . . . . 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 944 . . . . . . . . . . . . . 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 23980 . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 663 . . . . . . . . . . 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 601 . . . . . . . . . 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 1174 . . . . . . . 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 371 . . . . . . 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 1171 . . . 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 520 . . 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 808 . 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 250 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 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    \/ w3o 938    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   <.cop 3584   class class class wbr 3963   ` cfv 4638   NNcn 9679   EEcee 23856    Btwn cbtwn 23857    Colinear ccolin 24000  OutsideOfcoutsideof 24082
This theorem is referenced by:  outsidene1  24086  outsidene2  24087  btwnoutside  24088  broutsideof3  24089  outsideofcom  24091  outsideoftr  24092  outsideofeq  24093  outsideofeu  24094  lineunray  24110
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-oi 7158  df-card 7505  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-sum 12089  df-ee 23859  df-btwn 23860  df-cgr 23861  df-colinear 24004  df-outsideof 24083
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