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Theorem broutsideof2 32354
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⟩ ↔ (𝐴𝑃𝐵𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))))

Proof of Theorem broutsideof2
StepHypRef Expression
1 broutsideof 32353 . 2 (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))
2 btwntriv1 32248 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐴 Btwn ⟨𝐴, 𝐵⟩)
323adant3r1 1295 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐴 Btwn ⟨𝐴, 𝐵⟩)
4 breq1 4688 . . . . . . . 8 (𝐴 = 𝑃 → (𝐴 Btwn ⟨𝐴, 𝐵⟩ ↔ 𝑃 Btwn ⟨𝐴, 𝐵⟩))
53, 4syl5ibcom 235 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 = 𝑃𝑃 Btwn ⟨𝐴, 𝐵⟩))
65necon3bd 2837 . . . . . 6 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩ → 𝐴𝑃))
76imp 444 . . . . 5 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩) → 𝐴𝑃)
87adantrl 752 . . . 4 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝐴𝑃)
9 btwntriv2 32244 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐵 Btwn ⟨𝐴, 𝐵⟩)
1093adant3r1 1295 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐵 Btwn ⟨𝐴, 𝐵⟩)
11 breq1 4688 . . . . . . . 8 (𝐵 = 𝑃 → (𝐵 Btwn ⟨𝐴, 𝐵⟩ ↔ 𝑃 Btwn ⟨𝐴, 𝐵⟩))
1210, 11syl5ibcom 235 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐵 = 𝑃𝑃 Btwn ⟨𝐴, 𝐵⟩))
1312necon3bd 2837 . . . . . 6 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩ → 𝐵𝑃))
1413imp 444 . . . . 5 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩) → 𝐵𝑃)
1514adantrl 752 . . . 4 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝐵𝑃)
16 brcolinear 32291 . . . . . 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 pm2.24 121 . . . . . . . 8 (𝑃 Btwn ⟨𝐴, 𝐵⟩ → (¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩ → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩)))
1817a1i 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 1060 . . . . . . . . . 10 ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)))
20 btwncom 32246 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝑃⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝐵⟩))
2119, 20sylan2b 491 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝑃⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝐵⟩))
22 orc 399 . . . . . . . . 9 (𝐴 Btwn ⟨𝑃, 𝐵⟩ → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))
2321, 22syl6bi 243 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝑃⟩ → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩)))
2423a1dd 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 398 . . . . . . . . 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 ∈ 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⟩))))
2818, 24, 273jaod 1432 . . . . . 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⟩))))
2916, 28sylbid 230 . . . . 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⟩))))
3029imp32 448 . . . 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⟩))
318, 15, 303jca 1261 . . 3 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → (𝐴𝑃𝐵𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩)))
32 simp3 1083 . . . . . 6 ((𝐴𝑃𝐵𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩)) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))
33 3ancomb 1064 . . . . . . . 8 ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ↔ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)))
34 btwncolinear2 32302 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ → 𝑃 Colinear ⟨𝐴, 𝐵⟩))
3533, 34sylan2b 491 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ → 𝑃 Colinear ⟨𝐴, 𝐵⟩))
36 btwncolinear1 32301 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝑃, 𝐴⟩ → 𝑃 Colinear ⟨𝐴, 𝐵⟩))
3735, 36jaod 394 . . . . . 6 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩) → 𝑃 Colinear ⟨𝐴, 𝐵⟩))
3832, 37syl5 34 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝐴𝑃𝐵𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩)) → 𝑃 Colinear ⟨𝐴, 𝐵⟩))
3938imp 444 . . . 4 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐴𝑃𝐵𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))) → 𝑃 Colinear ⟨𝐴, 𝐵⟩)
40 simpr2 1088 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴𝑃𝐵𝑃)) → 𝐴𝑃)
4140neneqd 2828 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴𝑃𝐵𝑃)) → ¬ 𝐴 = 𝑃)
42 simprl1 1126 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴𝑃𝐵𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝐴 Btwn ⟨𝑃, 𝐵⟩)
43 simprr 811 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴𝑃𝐵𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝑃 Btwn ⟨𝐴, 𝐵⟩)
44 simpl 472 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
45 simpr2 1088 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
46 simpr1 1087 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝑃 ∈ (𝔼‘𝑁))
47 simpr3 1089 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
48 btwnswapid 32249 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩) → 𝐴 = 𝑃))
4944, 45, 46, 47, 48syl13anc 1368 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩) → 𝐴 = 𝑃))
5049adantr 480 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴𝑃𝐵𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩) → 𝐴 = 𝑃))
5142, 43, 50mp2and 715 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴𝑃𝐵𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝐴 = 𝑃)
5251expr 642 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴𝑃𝐵𝑃)) → (𝑃 Btwn ⟨𝐴, 𝐵⟩ → 𝐴 = 𝑃))
5341, 52mtod 189 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∧ 𝐴𝑃𝐵𝑃)) → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩)
54533exp2 1307 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ → (𝐴𝑃 → (𝐵𝑃 → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))))
55 simpr3 1089 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴𝑃𝐵𝑃)) → 𝐵𝑃)
5655neneqd 2828 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴𝑃𝐵𝑃)) → ¬ 𝐵 = 𝑃)
57 simprl1 1126 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴𝑃𝐵𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝐵 Btwn ⟨𝑃, 𝐴⟩)
58 simprr 811 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴𝑃𝐵𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝑃 Btwn ⟨𝐴, 𝐵⟩)
5944, 46, 45, 47, 58btwncomand 32247 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴𝑃𝐵𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝑃 Btwn ⟨𝐵, 𝐴⟩)
60 3anrot 1060 . . . . . . . . . . . . . 14 ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ↔ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)))
61 btwnswapid 32249 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝑃 Btwn ⟨𝐵, 𝐴⟩) → 𝐵 = 𝑃))
6260, 61sylan2br 492 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝑃 Btwn ⟨𝐵, 𝐴⟩) → 𝐵 = 𝑃))
6362adantr 480 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴𝑃𝐵𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝑃 Btwn ⟨𝐵, 𝐴⟩) → 𝐵 = 𝑃))
6457, 59, 63mp2and 715 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ ((𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴𝑃𝐵𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐵⟩)) → 𝐵 = 𝑃)
6564expr 642 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴𝑃𝐵𝑃)) → (𝑃 Btwn ⟨𝐴, 𝐵⟩ → 𝐵 = 𝑃))
6656, 65mtod 189 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐵 Btwn ⟨𝑃, 𝐴⟩ ∧ 𝐴𝑃𝐵𝑃)) → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩)
67663exp2 1307 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝑃, 𝐴⟩ → (𝐴𝑃 → (𝐵𝑃 → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))))
6854, 67jaod 394 . . . . . . 7 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩) → (𝐴𝑃 → (𝐵𝑃 → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))))
6968com12 32 . . . . . 6 ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩) → ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴𝑃 → (𝐵𝑃 → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))))
7069com4l 92 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴𝑃 → (𝐵𝑃 → ((𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩) → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))))
71703imp2 1304 . . . 4 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐴𝑃𝐵𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))) → ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩)
7239, 71jca 553 . . 3 (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ (𝐴𝑃𝐵𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))) → (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))
7331, 72impbida 895 . 2 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → ((𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩) ↔ (𝐴𝑃𝐵𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))))
741, 73syl5bb 272 1 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ (𝐴𝑃𝐵𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3o 1053  w3a 1054   = wceq 1523  wcel 2030  wne 2823  cop 4216   class class class wbr 4685  cfv 5926  cn 11058  𝔼cee 25813   Btwn cbtwn 25814   Colinear ccolin 32269  OutsideOfcoutsideof 32351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-ee 25816  df-btwn 25817  df-cgr 25818  df-colinear 32271  df-outsideof 32352
This theorem is referenced by:  outsidene1  32355  outsidene2  32356  btwnoutside  32357  broutsideof3  32358  outsideofcom  32360  outsideoftr  32361  outsideofeq  32362  outsideofeu  32363  lineunray  32379
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