Theorem outsidele 36420

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 485	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
2		simpr1 1204	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝑃 ∈ (𝔼‘𝑁))
3		simpr2 1205	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
4		simpr3 1206	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
5		brsegle2 36397	. . . . . 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 1390	. . . . 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 483	. . . 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 778	. . . . . . . . . 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 36416	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ 𝑃OutsideOf⟨𝐵, 𝐴⟩))
10	9	ad2antrr 734	. . . . . . . . . 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 234	. . . . . . . . 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 774	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
13		simplr1 1225	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑃 ∈ (𝔼‘𝑁))
14		simplr3 1227	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
15	12, 13, 14	cgrrflxd 36276	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → ⟨𝑃, 𝐵⟩Cgr⟨𝑃, 𝐵⟩)
16	15	adantr 483	. . . . . . . . 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 518	. . . . . . . 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 788	. . . . . . . . . . 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 487	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝑦 ∈ (𝔼‘𝑁))
20		simplr2 1226	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁))
21		btwncolinear1 36357	. . . . . . . . . . . . 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 1383	. . . . . . . . . . . 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 483	. . . . . . . . . . 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 36411	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ → 𝐴 ≠ 𝑃))
26	25	ad2antrr 734	. . . . . . . . . . . . 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 2952	. . . . . . . . . . 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 1097	. . . . . . . . . . . . 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 1242	. . . . . . . . . . . . . . 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 36303	. . . . . . . . . . . . . 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 1206	. . . . . . . . . . . . . 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 36306	. . . . . . . . . . . . . . . 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 1383	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 707	. . . . . . . . . . . . 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 603	. . . . . . . . . . . 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 459	. . . . . . . . . . 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 200	. . . . . . . . . 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 36409	. . . . . . . . . 10 ⊢ (𝑃OutsideOf⟨𝑦, 𝐴⟩ ↔ (𝑃 Colinear ⟨𝑦, 𝐴⟩ ∧ ¬ 𝑃 Btwn ⟨𝑦, 𝐴⟩))
41	24, 39, 40	sylanbrc 591	. . . . . . . . 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 789	. . . . . . . . 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 518	. . . . . . . 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 36418	. . . . . . . . . 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 1404	. . . . . . . . 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 483	. . . . . . . 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 707	. . . . . . 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 4822	. . . . . . . . 9 ⊢ (𝐵 = 𝑦 → ⟨𝑃, 𝐵⟩ = ⟨𝑃, 𝑦⟩)
49	48	breq2d 5102	. . . . . . . 8 ⊢ (𝐵 = 𝑦 → (𝐴 Btwn ⟨𝑃, 𝐵⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝑦⟩))
50	18, 49	syl5ibrcom 249	. . . . . . 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 668	. . . . 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 3154	. . . 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 242	. . 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 36405	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ → ⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩))
56	55	adantr 483	. . 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 214	. 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 415	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 208   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ∃wrex 3076  ⟨cop 4578   class class class wbr 5090  ‘cfv 6506  ℕcn 12196  𝔼cee 29023   Btwn cbtwn 29024  Cgrccgr 29025   Colinear ccolin 36325   Seg_≤ csegle 36394  OutsideOfcoutsideof 36407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-inf2 9582  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-isom 6515  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-er 8662  df-map 8794  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-sup 9374  df-oi 9444  df-card 9883  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-div 11831  df-nn 12197  df-2 12266  df-3 12267  df-n0 12468  df-z 12555  df-uz 12826  df-rp 12980  df-ico 13341  df-icc 13342  df-fz 13499  df-fzo 13646  df-seq 14001  df-exp 14061  df-hash 14330  df-cj 15098  df-re 15099  df-im 15100  df-sqrt 15234  df-abs 15235  df-clim 15487  df-sum 15686  df-ee 29026  df-btwn 29027  df-cgr 29028  df-ofs 36271  df-colinear 36327  df-ifs 36328  df-cgr3 36329  df-fs 36330  df-segle 36395  df-outsideof 36408

This theorem is referenced by: (None)