Theorem outsidele 36096

Description: Relate OutsideOf to Seg_≤. Theorem 6.13 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
outsidele	⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ → (⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝐵⟩)))

Proof of Theorem outsidele

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
2		simpr1 1194	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝑃 ∈ (𝔼‘𝑁))
3		simpr2 1195	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
4		simpr3 1196	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
5		brsegle2 36073	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩ ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩)))
6	1, 2, 3, 2, 4, 5	syl122anc 1379	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩ ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩)))
7	6	adantr 480	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑃OutsideOf⟨𝐴, 𝐵⟩) → (⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩ ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩)))
8		simprl 770	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → 𝑃OutsideOf⟨𝐴, 𝐵⟩)
9		outsideofcom 36092	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ 𝑃OutsideOf⟨𝐵, 𝐴⟩))
10	9	ad2antrr 725	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ 𝑃OutsideOf⟨𝐵, 𝐴⟩))
11	8, 10	mpbid 232	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → 𝑃OutsideOf⟨𝐵, 𝐴⟩)
12		simpll 766	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
13		simplr1 1215	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑃 ∈ (𝔼‘𝑁))
14		simplr3 1217	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
15	12, 13, 14	cgrrflxd 35952	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → ⟨𝑃, 𝐵⟩Cgr⟨𝑃, 𝐵⟩)
16	15	adantr 480	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → ⟨𝑃, 𝐵⟩Cgr⟨𝑃, 𝐵⟩)
17	11, 16	jca 511	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → (𝑃OutsideOf⟨𝐵, 𝐴⟩ ∧ ⟨𝑃, 𝐵⟩Cgr⟨𝑃, 𝐵⟩))
18		simprrl 780	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → 𝐴 Btwn ⟨𝑃, 𝑦⟩)
19		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑦 ∈ (𝔼‘𝑁))
20		simplr2 1216	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
21		btwncolinear1 36033	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝑃, 𝑦⟩ → 𝑃 Colinear ⟨𝑦, 𝐴⟩))
22	12, 13, 19, 20, 21	syl13anc 1372	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (𝐴 Btwn ⟨𝑃, 𝑦⟩ → 𝑃 Colinear ⟨𝑦, 𝐴⟩))
23	22	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → (𝐴 Btwn ⟨𝑃, 𝑦⟩ → 𝑃 Colinear ⟨𝑦, 𝐴⟩))
24	18, 23	mpd 15	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → 𝑃 Colinear ⟨𝑦, 𝐴⟩)
25		outsidene1 36087	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ → 𝐴 ≠ 𝑃))
26	25	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ → 𝐴 ≠ 𝑃))
27	8, 26	mpd 15	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → 𝐴 ≠ 𝑃)
28	27	neneqd 2951	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → ¬ 𝐴 = 𝑃)
29		df-3an 1089	. . . . . . . . . . . . 13 ⊢ ((𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩) ∧ 𝑃 Btwn ⟨𝑦, 𝐴⟩) ↔ ((𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩)) ∧ 𝑃 Btwn ⟨𝑦, 𝐴⟩))
30		simpr2l 1232	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩) ∧ 𝑃 Btwn ⟨𝑦, 𝐴⟩)) → 𝐴 Btwn ⟨𝑃, 𝑦⟩)
31	12, 20, 13, 19, 30	btwncomand 35979	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩) ∧ 𝑃 Btwn ⟨𝑦, 𝐴⟩)) → 𝐴 Btwn ⟨𝑦, 𝑃⟩)
32		simpr3 1196	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩) ∧ 𝑃 Btwn ⟨𝑦, 𝐴⟩)) → 𝑃 Btwn ⟨𝑦, 𝐴⟩)
33		btwnswapid2 35982	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → ((𝐴 Btwn ⟨𝑦, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑦, 𝐴⟩) → 𝐴 = 𝑃))
34	12, 20, 19, 13, 33	syl13anc 1372	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → ((𝐴 Btwn ⟨𝑦, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑦, 𝐴⟩) → 𝐴 = 𝑃))
35	34	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩) ∧ 𝑃 Btwn ⟨𝑦, 𝐴⟩)) → ((𝐴 Btwn ⟨𝑦, 𝑃⟩ ∧ 𝑃 Btwn ⟨𝑦, 𝐴⟩) → 𝐴 = 𝑃))
36	31, 32, 35	mp2and 698	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩) ∧ 𝑃 Btwn ⟨𝑦, 𝐴⟩)) → 𝐴 = 𝑃)
37	29, 36	sylan2br 594	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ ((𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩)) ∧ 𝑃 Btwn ⟨𝑦, 𝐴⟩)) → 𝐴 = 𝑃)
38	37	expr 456	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → (𝑃 Btwn ⟨𝑦, 𝐴⟩ → 𝐴 = 𝑃))
39	28, 38	mtod 198	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → ¬ 𝑃 Btwn ⟨𝑦, 𝐴⟩)
40		broutsideof 36085	. . . . . . . . . 10 ⊢ (𝑃OutsideOf⟨𝑦, 𝐴⟩ ↔ (𝑃 Colinear ⟨𝑦, 𝐴⟩ ∧ ¬ 𝑃 Btwn ⟨𝑦, 𝐴⟩))
41	24, 39, 40	sylanbrc 582	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → 𝑃OutsideOf⟨𝑦, 𝐴⟩)
42		simprrr 781	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩)
43	41, 42	jca 511	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → (𝑃OutsideOf⟨𝑦, 𝐴⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))
44		outsideofeq 36094	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁))) → (((𝑃OutsideOf⟨𝐵, 𝐴⟩ ∧ ⟨𝑃, 𝐵⟩Cgr⟨𝑃, 𝐵⟩) ∧ (𝑃OutsideOf⟨𝑦, 𝐴⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩)) → 𝐵 = 𝑦))
45	12, 13, 20, 13, 14, 14, 19, 44	syl133anc 1393	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (((𝑃OutsideOf⟨𝐵, 𝐴⟩ ∧ ⟨𝑃, 𝐵⟩Cgr⟨𝑃, 𝐵⟩) ∧ (𝑃OutsideOf⟨𝑦, 𝐴⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩)) → 𝐵 = 𝑦))
46	45	adantr 480	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → (((𝑃OutsideOf⟨𝐵, 𝐴⟩ ∧ ⟨𝑃, 𝐵⟩Cgr⟨𝑃, 𝐵⟩) ∧ (𝑃OutsideOf⟨𝑦, 𝐴⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩)) → 𝐵 = 𝑦))
47	17, 43, 46	mp2and 698	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → 𝐵 = 𝑦)
48		opeq2 4898	. . . . . . . . 9 ⊢ (𝐵 = 𝑦 → ⟨𝑃, 𝐵⟩ = ⟨𝑃, 𝑦⟩)
49	48	breq2d 5178	. . . . . . . 8 ⊢ (𝐵 = 𝑦 → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝑦⟩))
50	18, 49	syl5ibrcom 247	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → (𝐵 = 𝑦 → 𝐴 Btwn ⟨𝑃, 𝐵⟩))
51	47, 50	mpd 15	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → 𝐴 Btwn ⟨𝑃, 𝐵⟩)
52	51	an4s 659	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑃OutsideOf⟨𝐴, 𝐵⟩) ∧ (𝑦 ∈ (𝔼‘𝑁) ∧ (𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩))) → 𝐴 Btwn ⟨𝑃, 𝐵⟩)
53	52	rexlimdvaa 3162	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑃OutsideOf⟨𝐴, 𝐵⟩) → (∃𝑦 ∈ (𝔼‘𝑁)(𝐴 Btwn ⟨𝑃, 𝑦⟩ ∧ ⟨𝑃, 𝑦⟩Cgr⟨𝑃, 𝐵⟩) → 𝐴 Btwn ⟨𝑃, 𝐵⟩))
54	7, 53	sylbid 240	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑃OutsideOf⟨𝐴, 𝐵⟩) → (⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩ → 𝐴 Btwn ⟨𝑃, 𝐵⟩))
55		btwnsegle 36081	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ → ⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩))
56	55	adantr 480	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑃OutsideOf⟨𝐴, 𝐵⟩) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ → ⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩))
57	54, 56	impbid 212	. 2 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑃OutsideOf⟨𝐴, 𝐵⟩) → (⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝐵⟩))
58	57	ex 412	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ → (⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝐵⟩)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076  ⟨cop 4654   class class class wbr 5166  ‘cfv 6573  ℕcn 12293  𝔼cee 28921   Btwn cbtwn 28922  Cgrccgr 28923   Colinear ccolin 36001   Seg_≤ csegle 36070  OutsideOfcoutsideof 36083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-ee 28924  df-btwn 28925  df-cgr 28926  df-ofs 35947  df-colinear 36003  df-ifs 36004  df-cgr3 36005  df-fs 36006  df-segle 36071  df-outsideof 36084

This theorem is referenced by: (None)