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Theorem broutsideof2 28158
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 28157 . 2  |-  ( POutsideOf <. A ,  B >.  <->  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )
2 btwntriv1 28052 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  A  Btwn  <. A ,  B >. )
323adant3r1 1196 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  Btwn  <. A ,  B >. )
4 breq1 4300 . . . . . . . 8  |-  ( A  =  P  ->  ( A  Btwn  <. A ,  B >.  <-> 
P  Btwn  <. A ,  B >. ) )
53, 4syl5ibcom 220 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( A  =  P  ->  P  Btwn  <. A ,  B >. ) )
65necon3bd 2650 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( -.  P  Btwn  <. A ,  B >.  ->  A  =/=  P ) )
76imp 429 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  -.  P  Btwn  <. A ,  B >. )  ->  A  =/=  P )
87adantrl 715 . . . 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 28048 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  B  Btwn  <. A ,  B >. )
1093adant3r1 1196 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  Btwn  <. A ,  B >. )
11 breq1 4300 . . . . . . . 8  |-  ( B  =  P  ->  ( B  Btwn  <. A ,  B >.  <-> 
P  Btwn  <. A ,  B >. ) )
1210, 11syl5ibcom 220 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( B  =  P  ->  P  Btwn  <. A ,  B >. ) )
1312necon3bd 2650 . . . . . 6  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( -.  P  Btwn  <. A ,  B >.  ->  B  =/=  P ) )
1413imp 429 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  /\  -.  P  Btwn  <. A ,  B >. )  ->  B  =/=  P )
1514adantrl 715 . . . 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 28095 . . . . . 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 pm2.24 109 . . . . . . . 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 970 . . . . . . . . . 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 28050 . . . . . . . . . 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 475 . . . . . . . . 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 385 . . . . . . . . 9  |-  ( A 
Btwn  <. P ,  B >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
2321, 22syl6bi 228 . . . . . . . 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 46 . . . . . . 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 384 . . . . . . . . 9  |-  ( B 
Btwn  <. P ,  A >.  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
2625a1d 25 . . . . . . . 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 1282 . . . . . 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 215 . . . . 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 433 . . . 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 1168 . . 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 990 . . . . . 6  |-  ( ( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  -> 
( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
33 3ancomb 974 . . . . . . . 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 28106 . . . . . . . 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 475 . . . . . . 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 28105 . . . . . . 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 380 . . . . . 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 32 . . . . 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 429 . . . 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 995 . . . . . . . . . . 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 2629 . . . . . . . . . 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 1033 . . . . . . . . . . . 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 756 . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  N  e.  NN )
45 simpr2 995 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
46 simpr1 994 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  P  e.  ( EE `  N ) )
47 simpr3 996 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
48 btwnswapid 28053 . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 679 . . . . . . . . . . 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 615 . . . . . . . . . 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 177 . . . . . . . . 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 1205 . . . . . . . 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 996 . . . . . . . . . . 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 2629 . . . . . . . . . 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 1033 . . . . . . . . . . . 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 756 . . . . . . . . . . . . 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 28051 . . . . . . . . . . . 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 970 . . . . . . . . . . . . . 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 28053 . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 679 . . . . . . . . . . 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 615 . . . . . . . . . 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 177 . . . . . . . . 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 1205 . . . . . . . 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 380 . . . . . . 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 31 . . . . . 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 84 . . . . 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 1202 . . . 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 532 . . 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 828 . 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 257 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   <.cop 3888   class class class wbr 4297   ` cfv 5423   NNcn 10327   EEcee 23139    Btwn cbtwn 23140    Colinear ccolin 28073  OutsideOfcoutsideof 28155
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-ee 23142  df-btwn 23143  df-cgr 23144  df-colinear 28075  df-outsideof 28156
This theorem is referenced by:  outsidene1  28159  outsidene2  28160  btwnoutside  28161  broutsideof3  28162  outsideofcom  28164  outsideoftr  28165  outsideofeq  28166  outsideofeu  28167  lineunray  28183
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