Step |
Hyp |
Ref |
Expression |
1 |
|
broutsideof |
|- ( P OutsideOf <. A , B >. <-> ( P Colinear <. A , B >. /\ -. P Btwn <. A , B >. ) ) |
2 |
|
btwntriv1 |
|- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> A Btwn <. A , B >. ) |
3 |
2
|
3adant3r1 |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> A Btwn <. A , B >. ) |
4 |
|
breq1 |
|- ( A = P -> ( A Btwn <. A , B >. <-> P Btwn <. A , B >. ) ) |
5 |
3 4
|
syl5ibcom |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A = P -> P Btwn <. A , B >. ) ) |
6 |
5
|
necon3bd |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( -. P Btwn <. A , B >. -> A =/= P ) ) |
7 |
6
|
imp |
|- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ -. P Btwn <. A , B >. ) -> A =/= P ) |
8 |
7
|
adantrl |
|- ( ( ( 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 |
|- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> B Btwn <. A , B >. ) |
10 |
9
|
3adant3r1 |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> B Btwn <. A , B >. ) |
11 |
|
breq1 |
|- ( B = P -> ( B Btwn <. A , B >. <-> P Btwn <. A , B >. ) ) |
12 |
10 11
|
syl5ibcom |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( B = P -> P Btwn <. A , B >. ) ) |
13 |
12
|
necon3bd |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( -. P Btwn <. A , B >. -> B =/= P ) ) |
14 |
13
|
imp |
|- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ -. P Btwn <. A , B >. ) -> B =/= P ) |
15 |
14
|
adantrl |
|- ( ( ( 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 |
|- ( ( 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 |
|- ( P Btwn <. A , B >. -> ( -. P Btwn <. A , B >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) |
18 |
17
|
a1i |
|- ( ( 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 |
|- ( ( 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 |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) -> ( A Btwn <. B , P >. <-> A Btwn <. P , B >. ) ) |
21 |
19 20
|
sylan2b |
|- ( ( 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 |
|- ( A Btwn <. P , B >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) |
23 |
21 22
|
syl6bi |
|- ( ( 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 >. ) ) ) |
24 |
23
|
a1dd |
|- ( ( 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 |
|- ( B Btwn <. P , A >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) |
26 |
25
|
a1d |
|- ( B Btwn <. P , A >. -> ( -. P Btwn <. A , B >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) |
27 |
26
|
a1i |
|- ( ( 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 >. ) ) ) ) |
28 |
18 24 27
|
3jaod |
|- ( ( 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 >. ) ) ) ) |
29 |
16 28
|
sylbid |
|- ( ( 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 >. ) ) ) ) |
30 |
29
|
imp32 |
|- ( ( ( 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 >. ) ) |
31 |
8 15 30
|
3jca |
|- ( ( ( 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 |
|- ( ( A =/= P /\ B =/= P /\ ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) |
33 |
|
3ancomb |
|- ( ( 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 |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) -> ( A Btwn <. P , B >. -> P Colinear <. A , B >. ) ) |
35 |
33 34
|
sylan2b |
|- ( ( 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 |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( B Btwn <. P , A >. -> P Colinear <. A , B >. ) ) |
37 |
35 36
|
jaod |
|- ( ( 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 >. ) ) |
38 |
32 37
|
syl5 |
|- ( ( 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 >. ) ) |
39 |
38
|
imp |
|- ( ( ( 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 |
|- ( ( ( 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 ) |
41 |
40
|
neneqd |
|- ( ( ( 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 |
|- ( ( ( 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 |
|- ( ( ( 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 |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> N e. NN ) |
45 |
|
simpr2 |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> A e. ( EE ` N ) ) |
46 |
|
simpr1 |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> P e. ( EE ` N ) ) |
47 |
|
simpr3 |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> B e. ( EE ` N ) ) |
48 |
|
btwnswapid |
|- ( ( 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 ) ) |
49 |
44 45 46 47 48
|
syl13anc |
|- ( ( 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 ) ) |
50 |
49
|
adantr |
|- ( ( ( 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 ) ) |
51 |
42 43 50
|
mp2and |
|- ( ( ( 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 ) |
52 |
51
|
expr |
|- ( ( ( 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 ) ) |
53 |
41 52
|
mtod |
|- ( ( ( 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 >. ) |
54 |
53
|
3exp2 |
|- ( ( 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 |
|- ( ( ( 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 ) |
56 |
55
|
neneqd |
|- ( ( ( 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 |
|- ( ( ( 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 |
|- ( ( ( 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 >. ) |
59 |
44 46 45 47 58
|
btwncomand |
|- ( ( ( 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 |
|- ( ( 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 |
|- ( ( 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 ) ) |
62 |
60 61
|
sylan2br |
|- ( ( 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 ) ) |
63 |
62
|
adantr |
|- ( ( ( 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 ) ) |
64 |
57 59 63
|
mp2and |
|- ( ( ( 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 ) |
65 |
64
|
expr |
|- ( ( ( 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 ) ) |
66 |
56 65
|
mtod |
|- ( ( ( 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 >. ) |
67 |
66
|
3exp2 |
|- ( ( 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 >. ) ) ) ) |
68 |
54 67
|
jaod |
|- ( ( 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 >. ) ) ) ) |
69 |
68
|
com12 |
|- ( ( 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 >. ) ) ) ) |
70 |
69
|
com4l |
|- ( ( 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 >. ) ) ) ) |
71 |
70
|
3imp2 |
|- ( ( ( 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 >. ) |
72 |
39 71
|
jca |
|- ( ( ( 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 >. ) ) |
73 |
31 72
|
impbida |
|- ( ( 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 >. ) ) ) ) |
74 |
1 73
|
syl5bb |
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( P OutsideOf <. A , B >. <-> ( A =/= P /\ B =/= P /\ ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) ) |