Theorem outsidele 36624

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 487	. . . . . 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 36601	. . . . . 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 485	. . . 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 782	. . . . . . . . . 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 36620	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ 𝑃OutsideOf⟨𝐵, 𝐴⟩))
10	9	ad2antrr 738	. . . . . . . . . 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 778	. . . . . . . . . . 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 36480	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) ∧ 𝑦 ∈ (𝔼‘𝑁)) → ⟨𝑃, 𝐵⟩Cgr⟨𝑃, 𝐵⟩)
16	15	adantr 485	. . . . . . . . 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 520	. . . . . . . 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 792	. . . . . . . . . . 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 489	. . . . . . . . . . . . 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 36561	. . . . . . . . . . . . 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 485	. . . . . . . . . . 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 36615	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ → 𝐴 ≠ 𝑃))
26	25	ad2antrr 738	. . . . . . . . . . . . 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 2963	. . . . . . . . . . 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 36507	. . . . . . . . . . . . . 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 36510	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 711	. . . . . . . . . . . . 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 606	. . . . . . . . . . . 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 461	. . . . . . . . . . 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 36613	. . . . . . . . . 10 ⊢ (𝑃OutsideOf⟨𝑦, 𝐴⟩ ↔ (𝑃 Colinear ⟨𝑦, 𝐴⟩ ∧ ¬ 𝑃 Btwn ⟨𝑦, 𝐴⟩))
41	24, 39, 40	sylanbrc 594	. . . . . . . . 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 793	. . . . . . . . 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 520	. . . . . . . 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 36622	. . . . . . . . . 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 485	. . . . . . . 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 711	. . . . . . 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 4839	. . . . . . . . 9 ⊢ (𝐵 = 𝑦 → ⟨𝑃, 𝐵⟩ = ⟨𝑃, 𝑦⟩)
49	48	breq2d 5121	. . . . . . . 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 672	. . . . 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 3167	. . . 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 36609	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝑃, 𝐵⟩ → ⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩))
56	55	adantr 485	. . 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 417	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 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089  ⟨cop 4595   class class class wbr 5109  ‘cfv 6536  ℕcn 12228  𝔼cee 29237   Btwn cbtwn 29238  Cgrccgr 29239   Colinear ccolin 36529   Seg_≤ csegle 36598  OutsideOfcoutsideof 36611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-n0 12500  df-z 12587  df-uz 12858  df-rp 13012  df-ico 13373  df-icc 13374  df-fz 13531  df-fzo 13679  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-sum 15734  df-ee 29240  df-btwn 29241  df-cgr 29242  df-ofs 36475  df-colinear 36531  df-ifs 36532  df-cgr3 36533  df-fs 36534  df-segle 36599  df-outsideof 36612

This theorem is referenced by: (None)