Theorem outsidele 36897

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 488	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
2		simpr1 1213	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝑃 ∈ (𝔼‘𝑁))
3		simpr2 1214	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
4		simpr3 1215	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
5		brsegle2 36874	. . . . . 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 1406	. . . . 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 486	. . . 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 783	. . . . . . . . . 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 36893	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ 𝑃OutsideOf⟨𝐵, 𝐴⟩))
10	9	ad2antrr 739	. . . . . . . . . 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 235	. . . . . . . . 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 779	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
13		simplr1 1234	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑃 ∈ (𝔼‘𝑁))
14		simplr3 1236	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
15	12, 13, 14	cgrrflxd 36753	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → ⟨𝑃, 𝐵⟩Cgr⟨𝑃, 𝐵⟩)
16	15	adantr 486	. . . . . . . . 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 521	. . . . . . . 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 793	. . . . . . . . . . 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 490	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑦 ∈ (𝔼‘𝑁))
20		simplr2 1235	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
21		btwncolinear1 36834	. . . . . . . . . . . . 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 1399	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 16	. . . . . . . . . 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 36888	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ → 𝐴 ≠ 𝑃))
26	25	ad2antrr 739	. . . . . . . . . . . . 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 16	. . . . . . . . . . . 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 2961	. . . . . . . . . . 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 1105	. . . . . . . . . . . . 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 1251	. . . . . . . . . . . . . . 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 36780	. . . . . . . . . . . . . 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 1215	. . . . . . . . . . . . . 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 36783	. . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 712	. . . . . . . . . . . . 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 607	. . . . . . . . . . . 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 462	. . . . . . . . . . 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 201	. . . . . . . . . 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 36886	. . . . . . . . . 10 ⊢ (𝑃OutsideOf⟨𝑦, 𝐴⟩ ↔ (𝑃 Colinear ⟨𝑦, 𝐴⟩ ∧ ¬ 𝑃 Btwn ⟨𝑦, 𝐴⟩))
41	24, 39, 40	sylanbrc 595	. . . . . . . . 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 794	. . . . . . . . 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 521	. . . . . . . 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 36895	. . . . . . . . . 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 1420	. . . . . . . . 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 486	. . . . . . . 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 712	. . . . . . 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 4834	. . . . . . . . 9 ⊢ (𝐵 = 𝑦 → ⟨𝑃, 𝐵⟩ = ⟨𝑃, 𝑦⟩)
49	48	breq2d 5115	. . . . . . . 8 ⊢ (𝐵 = 𝑦 → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝑦⟩))
50	18, 49	syl5ibrcom 250	. . . . . . 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 16	. . . . . 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 673	. . . . 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 3165	. . . 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 243	. . 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 36882	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ → ⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩))
56	55	adantr 486	. . 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 215	. 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 418	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ → (⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝐵⟩)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087  ⟨cop 4590   class class class wbr 5103  ‘cfv 6538  ℕcn 12335  𝔼cee 29465   Btwn cbtwn 29466  Cgrccgr 29467   Colinear ccolin 36802   Seg_≤ csegle 36871  OutsideOfcoutsideof 36884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-inf2 9642  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  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 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-oi 9504  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-n0 12607  df-z 12694  df-uz 12966  df-rp 13121  df-ico 13482  df-icc 13483  df-fz 13640  df-fzo 13789  df-seq 14145  df-exp 14205  df-hash 14475  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-clim 15655  df-sum 15854  df-ee 29468  df-btwn 29469  df-cgr 29470  df-ofs 36748  df-colinear 36804  df-ifs 36805  df-cgr3 36806  df-fs 36807  df-segle 36872  df-outsideof 36885

This theorem is used by: (None)