Theorem broutsideof2 36477

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 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))))

Proof of Theorem broutsideof2

Step	Hyp	Ref	Expression
1		broutsideof 36476	. 2 ⊢ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))
2		btwntriv1 36371	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐴 Btwn ⟨𝐴, 𝐵⟩)
3	2	3adant3r1 1197	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐴 Btwn ⟨𝐴, 𝐵⟩)
4		breq1 5105	. . . . . . . 8 ⊢ (𝐴 = 𝑃 → (𝐴 Btwn ⟨𝐴, 𝐵⟩ ↔ 𝑃 Btwn ⟨𝐴, 𝐵⟩))
5	3, 4	syl5ibcom 247	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 = 𝑃 → 𝑃 Btwn ⟨𝐴, 𝐵⟩))
6	5	necon3bd 2973	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩ → 𝐴 ≠ 𝑃))
7	6	imp 410	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩) → 𝐴 ≠ 𝑃)
8	7	adantrl 726	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝐴 ≠ 𝑃)
9		btwntriv2 36367	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐵 Btwn ⟨𝐴, 𝐵⟩)
10	9	3adant3r1 1197	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐵 Btwn ⟨𝐴, 𝐵⟩)
11		breq1 5105	. . . . . . . 8 ⊢ (𝐵 = 𝑃 → (𝐵 Btwn ⟨𝐴, 𝐵⟩ ↔ 𝑃 Btwn ⟨𝐴, 𝐵⟩))
12	10, 11	syl5ibcom 247	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐵 = 𝑃 → 𝑃 Btwn ⟨𝐴, 𝐵⟩))
13	12	necon3bd 2973	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩ → 𝐵 ≠ 𝑃))
14	13	imp 410	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩) → 𝐵 ≠ 𝑃)
15	14	adantrl 726	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝐵 ≠ 𝑃)
16		brcolinear 36414	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃 Colinear ⟨𝐴, 𝐵⟩ ↔ (𝑃 Btwn ⟨𝐴, 𝐵⟩ ∨ 𝐴 Btwn ⟨𝐵, 𝑃⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩)))
17		pm2.24 124	. . . . . . . 8 ⊢ (𝑃 Btwn ⟨𝐴, 𝐵⟩ → (¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩ → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩)))
18	17	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃 Btwn ⟨𝐴, 𝐵⟩ → (¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩ → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))))
19		3anrot 1113	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)))
20		btwncom 36369	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝑃⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝐵⟩))
21	19, 20	sylan2b 603	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝑃⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝐵⟩))
22		orc 878	. . . . . . . . 9 ⊢ (𝐴 Btwn ⟨𝑃, 𝐵⟩ → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))
23	21, 22	biimtrdi 255	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝑃⟩ → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩)))
24	23	a1dd 50	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝑃⟩ → (¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩ → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))))
25		olc 879	. . . . . . . . 9 ⊢ (𝐵 Btwn ⟨𝑃, 𝐴⟩ → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))
26	25	a1d 25	. . . . . . . 8 ⊢ (𝐵 Btwn ⟨𝑃, 𝐴⟩ → (¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩ → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩)))
27	26	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝑃, 𝐴⟩ → (¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩ → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))))
28	18, 24, 27	3jaod 1451	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘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 242	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃 Colinear ⟨𝐴, 𝐵⟩ → (¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩ → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))))
30	29	imp32 422	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))
31	8, 15, 30	3jca 1142	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → (𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩)))
32		simp3 1152	. . . . . 6 ⊢ ((𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩)) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))
33		3ancomb 1112	. . . . . . . 8 ⊢ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ↔ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)))
34		btwncolinear2 36425	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ → 𝑃 Colinear ⟨𝐴, 𝐵⟩))
35	33, 34	sylan2b 603	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ → 𝑃 Colinear ⟨𝐴, 𝐵⟩))
36		btwncolinear1 36424	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝑃, 𝐴⟩ → 𝑃 Colinear ⟨𝐴, 𝐵⟩))
37	35, 36	jaod 870	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩) → 𝑃 Colinear ⟨𝐴, 𝐵⟩))
38	32, 37	syl5 34	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩)) → 𝑃 Colinear ⟨𝐴, 𝐵⟩))
39	38	imp 410	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))) → 𝑃 Colinear ⟨𝐴, 𝐵⟩)
40		simpr2 1210	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃)) → 𝐴 ≠ 𝑃)
41	40	neneqd 2964	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃)) → ¬ 𝐴 = 𝑃)
42		simprl1 1233	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝐴 Btwn ⟨𝑃, 𝐵⟩)
43		simprr 782	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝑃 Btwn ⟨𝐴, 𝐵⟩)
44		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
45		simpr2 1210	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
46		simpr1 1209	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝑃 ∈ (𝔼‘𝑁))
47		simpr3 1211	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
48		btwnswapid 36372	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩) → 𝐴 = 𝑃))
49	44, 45, 46, 47, 48	syl13anc 1393	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩) → 𝐴 = 𝑃))
50	49	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘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 709	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝐴 = 𝑃)
52	51	expr 460	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃)) → (𝑃 Btwn ⟨𝐴, 𝐵⟩ → 𝐴 = 𝑃))
53	41, 52	mtod 200	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃)) → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩)
54	53	3exp2 1369	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ → (𝐴 ≠ 𝑃 → (𝐵 ≠ 𝑃 → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))))
55		simpr3 1211	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃)) → 𝐵 ≠ 𝑃)
56	55	neneqd 2964	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃)) → ¬ 𝐵 = 𝑃)
57		simprl1 1233	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝐵 Btwn ⟨𝑃, 𝐴⟩)
58		simprr 782	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝑃 Btwn ⟨𝐴, 𝐵⟩)
59	44, 46, 45, 47, 58	btwncomand 36370	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝑃 Btwn ⟨𝐵, 𝐴⟩)
60		3anrot 1113	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ↔ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)))
61		btwnswapid 36372	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝑃 Btwn ⟨𝐵, 𝐴⟩) → 𝐵 = 𝑃))
62	60, 61	sylan2br 604	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝑃 Btwn ⟨𝐵, 𝐴⟩) → 𝐵 = 𝑃))
63	62	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘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 709	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝐵 = 𝑃)
65	64	expr 460	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃)) → (𝑃 Btwn ⟨𝐴, 𝐵⟩ → 𝐵 = 𝑃))
66	56, 65	mtod 200	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃)) → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩)
67	66	3exp2 1369	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝑃, 𝐴⟩ → (𝐴 ≠ 𝑃 → (𝐵 ≠ 𝑃 → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))))
68	54, 67	jaod 870	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩) → (𝐴 ≠ 𝑃 → (𝐵 ≠ 𝑃 → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))))
69	68	com12 32	. . . . . 6 ⊢ ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩) → ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 ≠ 𝑃 → (𝐵 ≠ 𝑃 → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))))
70	69	com4l 92	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 ≠ 𝑃 → (𝐵 ≠ 𝑃 → ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩) → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))))
71	70	3imp2 1364	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))) → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩)
72	39, 71	jca 519	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘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 810	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩) ↔ (𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))))
74	1, 73	bitrid 285	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ (𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∨ w3o 1098   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ⟨cop 4590   class class class wbr 5102  ‘cfv 6523  ℕcn 12212  𝔼cee 29090   Btwn cbtwn 29091   Colinear ccolin 36392  OutsideOfcoutsideof 36474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-inf2 9598  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-er 8680  df-map 8812  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-sup 9390  df-oi 9460  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-div 11847  df-nn 12213  df-2 12282  df-3 12283  df-n0 12484  df-z 12571  df-uz 12842  df-rp 12996  df-ico 13357  df-icc 13358  df-fz 13515  df-fzo 13662  df-seq 14017  df-exp 14077  df-hash 14346  df-cj 15128  df-re 15129  df-im 15130  df-sqrt 15264  df-abs 15265  df-clim 15517  df-sum 15716  df-ee 29093  df-btwn 29094  df-cgr 29095  df-colinear 36394  df-outsideof 36475

This theorem is referenced by:  outsidene1  36478  outsidene2  36479  btwnoutside  36480  broutsideof3  36481  outsideofcom  36483  outsideoftr  36484  outsideofeq  36485  outsideofeu  36486  lineunray  36502