Theorem outsidele 36150

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 1195	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝑃 ∈ (𝔼‘𝑁))
3		simpr2 1196	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
4		simpr3 1197	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
5		brsegle2 36127	. . . . . 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 1381	. . . . 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 36146	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ 𝑃OutsideOf⟨𝐵, 𝐴⟩))
10	9	ad2antrr 726	. . . . . . . . . 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 1216	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑃 ∈ (𝔼‘𝑁))
14		simplr3 1218	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
15	12, 13, 14	cgrrflxd 36006	. . . . . . . . . 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 1217	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
21		btwncolinear1 36087	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 36141	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ → 𝐴 ≠ 𝑃))
26	25	ad2antrr 726	. . . . . . . . . . . . 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 2937	. . . . . . . . . . 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 1088	. . . . . . . . . . . . 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 1233	. . . . . . . . . . . . . . 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 36033	. . . . . . . . . . . . . 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 1197	. . . . . . . . . . . . . 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 36036	. . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . 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 699	. . . . . . . . . . . . 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 595	. . . . . . . . . . . 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 36139	. . . . . . . . . 10 ⊢ (𝑃OutsideOf⟨𝑦, 𝐴⟩ ↔ (𝑃 Colinear ⟨𝑦, 𝐴⟩ ∧ ¬ 𝑃 Btwn ⟨𝑦, 𝐴⟩))
41	24, 39, 40	sylanbrc 583	. . . . . . . . 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 36148	. . . . . . . . . 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 1395	. . . . . . . . 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 699	. . . . . . 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 4850	. . . . . . . . 9 ⊢ (𝐵 = 𝑦 → ⟨𝑃, 𝐵⟩ = ⟨𝑃, 𝑦⟩)
49	48	breq2d 5131	. . . . . . . 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 660	. . . . 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 3142	. . . 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 36135	. . . 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 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∃wrex 3060  ⟨cop 4607   class class class wbr 5119  ‘cfv 6531  ℕcn 12240  𝔼cee 28867   Btwn cbtwn 28868  Cgrccgr 28869   Colinear ccolin 36055   Seg_≤ csegle 36124  OutsideOfcoutsideof 36137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-sup 9454  df-oi 9524  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-3 12304  df-n0 12502  df-z 12589  df-uz 12853  df-rp 13009  df-ico 13368  df-icc 13369  df-fz 13525  df-fzo 13672  df-seq 14020  df-exp 14080  df-hash 14349  df-cj 15118  df-re 15119  df-im 15120  df-sqrt 15254  df-abs 15255  df-clim 15504  df-sum 15703  df-ee 28870  df-btwn 28871  df-cgr 28872  df-ofs 36001  df-colinear 36057  df-ifs 36058  df-cgr3 36059  df-fs 36060  df-segle 36125  df-outsideof 36138

This theorem is referenced by: (None)