lineunray - Mathbox for Scott Fenton

	Mathbox for Scott Fenton		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lineunray		Structured version Visualization version GIF version

Theorem lineunray 36593

Description: A line is composed of a point and the two rays emerging from it. Theorem 6.15 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
lineunray	⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → (𝑃 Btwn ⟨𝑄, 𝑅⟩ → (𝑃Line𝑄) = (((𝑃Ray𝑄) ∪ {𝑃}) ∪ (𝑃Ray𝑅))))

Proof of Theorem lineunray Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1208	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
2		simpr 489	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (𝔼‘𝑁))
3		simpl21 1268	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ∈ (𝔼‘𝑁))
4		simpl22 1269	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑄 ∈ (𝔼‘𝑁))
5		brcolinear 36505	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁))) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ (𝑥 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑄 Btwn ⟨𝑥, 𝑃⟩)))
6	1, 2, 3, 4, 5	syl13anc 1397	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ (𝑥 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑄 Btwn ⟨𝑥, 𝑃⟩)))
7	6	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ (𝑥 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑄 Btwn ⟨𝑥, 𝑃⟩)))
8		olc 881	. . . . . . . . . . . . . 14 ⊢ (𝑥 Btwn ⟨𝑃, 𝑄⟩ → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))
9	8	orcd 886	. . . . . . . . . . . . 13 ⊢ (𝑥 Btwn ⟨𝑃, 𝑄⟩ → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))
10	9	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑥 Btwn ⟨𝑃, 𝑄⟩ → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
11		simpl3l 1245	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ≠ 𝑄)
12	11	necomd 3011	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑄 ≠ 𝑃)
13	12	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑄 ≠ 𝑃)
14		simprl 782	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑃 Btwn ⟨𝑄, 𝑅⟩)
15		simprr 784	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → 𝑃 Btwn ⟨𝑄, 𝑥⟩)
16	13, 14, 15	3jca 1144	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → (𝑄 ≠ 𝑃 ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩))
17		simpl23 1270	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑅 ∈ (𝔼‘𝑁))
18		btwnconn2 36548	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)) ∧ (𝑅 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → ((𝑄 ≠ 𝑃 ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩) → (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))
19	1, 4, 3, 17, 2, 18	syl122anc 1404	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑄 ≠ 𝑃 ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩) → (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))
20	19	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → ((𝑄 ≠ 𝑃 ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩) → (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))
21	16, 20	mpd 16	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))
22	21	olcd 887	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑃 Btwn ⟨𝑄, 𝑥⟩)) → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))
23	22	expr 461	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑃 Btwn ⟨𝑄, 𝑥⟩ → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
24		btwncom 36460	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → (𝑄 Btwn ⟨𝑥, 𝑃⟩ ↔ 𝑄 Btwn ⟨𝑃, 𝑥⟩))
25	1, 4, 2, 3, 24	syl13anc 1397	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑄 Btwn ⟨𝑥, 𝑃⟩ ↔ 𝑄 Btwn ⟨𝑃, 𝑥⟩))
26		orc 880	. . . . . . . . . . . . . . 15 ⊢ (𝑄 Btwn ⟨𝑃, 𝑥⟩ → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))
27	26	orcd 886	. . . . . . . . . . . . . 14 ⊢ (𝑄 Btwn ⟨𝑃, 𝑥⟩ → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))
28	25, 27	biimtrdi 256	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑄 Btwn ⟨𝑥, 𝑃⟩ → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
29	28	adantr 485	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑄 Btwn ⟨𝑥, 𝑃⟩ → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
30	10, 23, 29	3jaod 1454	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → ((𝑥 Btwn ⟨𝑃, 𝑄⟩ ∨ 𝑃 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑄 Btwn ⟨𝑥, 𝑃⟩) → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
31	7, 30	sylbid 243	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
32		olc 881	. . . . . . . . . 10 ⊢ (((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)) → (𝑥 = 𝑃 ∨ ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
33	31, 32	syl6 36	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ → (𝑥 = 𝑃 ∨ ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))))
34		colineartriv1 36513	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁)) → 𝑃 Colinear ⟨𝑃, 𝑄⟩)
35	1, 3, 4, 34	syl3anc 1396	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 Colinear ⟨𝑃, 𝑄⟩)
36		breq1 5111	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑃 → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ 𝑃 Colinear ⟨𝑃, 𝑄⟩))
37	35, 36	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑥 = 𝑃 → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
38	37	adantr 485	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑥 = 𝑃 → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
39		btwncolinear3 36517	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁))) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
40	1, 3, 2, 4, 39	syl13anc 1397	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
41		btwncolinear5 36519	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → (𝑥 Btwn ⟨𝑃, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
42	1, 3, 4, 2, 41	syl13anc 1397	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑥 Btwn ⟨𝑃, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
43	40, 42	jaod 872	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
44	43	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
45		simpl3r 1246	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ≠ 𝑅)
46	45	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝑃, 𝑥⟩)) → 𝑃 ≠ 𝑅)
47		simprl 782	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝑃, 𝑥⟩)) → 𝑃 Btwn ⟨𝑄, 𝑅⟩)
48		simprr 784	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝑃, 𝑥⟩)) → 𝑅 Btwn ⟨𝑃, 𝑥⟩)
49	46, 47, 48	3jca 1144	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝑃, 𝑥⟩)) → (𝑃 ≠ 𝑅 ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝑃, 𝑥⟩))
50		btwnouttr 36470	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁)) ∧ (𝑅 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → ((𝑃 ≠ 𝑅 ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝑃, 𝑥⟩) → 𝑃 Btwn ⟨𝑄, 𝑥⟩))
51	1, 4, 3, 17, 2, 50	syl122anc 1404	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑃 ≠ 𝑅 ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝑃, 𝑥⟩) → 𝑃 Btwn ⟨𝑄, 𝑥⟩))
52	51	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝑃, 𝑥⟩)) → ((𝑃 ≠ 𝑅 ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝑃, 𝑥⟩) → 𝑃 Btwn ⟨𝑄, 𝑥⟩))
53	49, 52	mpd 16	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝑃, 𝑥⟩)) → 𝑃 Btwn ⟨𝑄, 𝑥⟩)
54		btwncolinear4 36518	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → (𝑃 Btwn ⟨𝑄, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
55	1, 4, 2, 3, 54	syl13anc 1397	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑃 Btwn ⟨𝑄, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
56	55	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝑃, 𝑥⟩)) → (𝑃 Btwn ⟨𝑄, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
57	53, 56	mpd 16	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝑃, 𝑥⟩)) → 𝑥 Colinear ⟨𝑃, 𝑄⟩)
58	57	expr 461	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑅 Btwn ⟨𝑃, 𝑥⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
59		simprr 784	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑅⟩)) → 𝑥 Btwn ⟨𝑃, 𝑅⟩)
60	1, 2, 3, 17, 59	btwncomand 36461	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑅⟩)) → 𝑥 Btwn ⟨𝑅, 𝑃⟩)
61		simprl 782	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑅⟩)) → 𝑃 Btwn ⟨𝑄, 𝑅⟩)
62	1, 3, 4, 17, 61	btwncomand 36461	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑅⟩)) → 𝑃 Btwn ⟨𝑅, 𝑄⟩)
63	1, 17, 2, 3, 4, 60, 62	btwnexch3and 36467	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑅⟩)) → 𝑃 Btwn ⟨𝑥, 𝑄⟩)
64		btwncolinear2 36516	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → (𝑃 Btwn ⟨𝑥, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
65	1, 2, 4, 3, 64	syl13anc 1397	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑃 Btwn ⟨𝑥, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
66	65	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑅⟩)) → (𝑃 Btwn ⟨𝑥, 𝑄⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
67	63, 66	mpd 16	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ (𝑃 Btwn ⟨𝑄, 𝑅⟩ ∧ 𝑥 Btwn ⟨𝑃, 𝑅⟩)) → 𝑥 Colinear ⟨𝑃, 𝑄⟩)
68	67	expr 461	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑥 Btwn ⟨𝑃, 𝑅⟩ → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
69	58, 68	jaod 872	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → ((𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩) → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
70	44, 69	jaod 872	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)) → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
71	38, 70	jaod 872	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → ((𝑥 = 𝑃 ∨ ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))) → 𝑥 Colinear ⟨𝑃, 𝑄⟩))
72	33, 71	impbid 215	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ (𝑥 = 𝑃 ∨ ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))))
73		pm5.63 1035	. . . . . . . . 9 ⊢ ((𝑥 = 𝑃 ∨ ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))) ↔ (𝑥 = 𝑃 ∨ (¬ 𝑥 = 𝑃 ∧ ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))))
74		df-ne 2957	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ 𝑃 ↔ ¬ 𝑥 = 𝑃)
75	74	anbi1i 635	. . . . . . . . . . 11 ⊢ ((𝑥 ≠ 𝑃 ∧ ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))) ↔ (¬ 𝑥 = 𝑃 ∧ ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
76		andi 1023	. . . . . . . . . . 11 ⊢ ((𝑥 ≠ 𝑃 ∧ ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))) ↔ ((𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) ∨ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
77	75, 76	bitr3i 280	. . . . . . . . . 10 ⊢ ((¬ 𝑥 = 𝑃 ∧ ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))) ↔ ((𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) ∨ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
78	77	orbi2i 925	. . . . . . . . 9 ⊢ ((𝑥 = 𝑃 ∨ (¬ 𝑥 = 𝑃 ∧ ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))) ↔ (𝑥 = 𝑃 ∨ ((𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) ∨ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))))
79	73, 78	bitri 278	. . . . . . . 8 ⊢ ((𝑥 = 𝑃 ∨ ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))) ↔ (𝑥 = 𝑃 ∨ ((𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) ∨ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))))
80	72, 79	bitrdi 290	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ (𝑥 = 𝑃 ∨ ((𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) ∨ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))))
81		broutsideof2 36568	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝑄, 𝑥⟩ ↔ (𝑄 ≠ 𝑃 ∧ 𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))))
82	1, 3, 4, 2, 81	syl13anc 1397	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑃OutsideOf⟨𝑄, 𝑥⟩ ↔ (𝑄 ≠ 𝑃 ∧ 𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))))
83		3simpc 1166	. . . . . . . . . . . 12 ⊢ ((𝑄 ≠ 𝑃 ∧ 𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → (𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))
84		simpl3l 1245	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ (𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))) → 𝑃 ≠ 𝑄)
85	84	necomd 3011	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ (𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))) → 𝑄 ≠ 𝑃)
86		simprrl 792	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ (𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))) → 𝑥 ≠ 𝑃)
87		simprrr 793	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ (𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))) → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))
88	85, 86, 87	3jca 1144	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ (𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))) → (𝑄 ≠ 𝑃 ∧ 𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)))
89	88	expr 461	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) → (𝑄 ≠ 𝑃 ∧ 𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))))
90	83, 89	impbid2 229	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑄 ≠ 𝑃 ∧ 𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) ↔ (𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))))
91	82, 90	bitrd 282	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑃OutsideOf⟨𝑄, 𝑥⟩ ↔ (𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))))
92		broutsideof2 36568	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝑅, 𝑥⟩ ↔ (𝑅 ≠ 𝑃 ∧ 𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
93	1, 3, 17, 2, 92	syl13anc 1397	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑃OutsideOf⟨𝑅, 𝑥⟩ ↔ (𝑅 ≠ 𝑃 ∧ 𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
94		3simpc 1166	. . . . . . . . . . . 12 ⊢ ((𝑅 ≠ 𝑃 ∧ 𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)) → (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))
95		simpl3r 1246	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))) → 𝑃 ≠ 𝑅)
96	95	necomd 3011	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))) → 𝑅 ≠ 𝑃)
97		simprrl 792	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))) → 𝑥 ≠ 𝑃)
98		simprrr 793	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))) → (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))
99	96, 97, 98	3jca 1144	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))) → (𝑅 ≠ 𝑃 ∧ 𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))
100	99	expr 461	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)) → (𝑅 ≠ 𝑃 ∧ 𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
101	94, 100	impbid2 229	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑅 ≠ 𝑃 ∧ 𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)) ↔ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
102	93, 101	bitrd 282	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑃OutsideOf⟨𝑅, 𝑥⟩ ↔ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))
103	91, 102	orbi12d 931	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩) ↔ ((𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) ∨ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))))
104	103	adantr 485	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩) ↔ ((𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) ∨ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))))
105	104	orbi2d 928	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → ((𝑥 = 𝑃 ∨ (𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩)) ↔ (𝑥 = 𝑃 ∨ ((𝑥 ≠ 𝑃 ∧ (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩)) ∨ (𝑥 ≠ 𝑃 ∧ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩))))))
106	80, 105	bitr4d 285	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ (𝑥 = 𝑃 ∨ (𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩))))
107		orcom 883	. . . . . . 7 ⊢ ((𝑥 = 𝑃 ∨ (𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩)) ↔ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩) ∨ 𝑥 = 𝑃))
108		or32 938	. . . . . . 7 ⊢ (((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩) ∨ 𝑥 = 𝑃) ↔ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃) ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩))
109	107, 108	bitri 278	. . . . . 6 ⊢ ((𝑥 = 𝑃 ∨ (𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩)) ↔ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃) ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩))
110	106, 109	bitrdi 290	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃) ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩)))
111	110	an32s 664	. . . 4 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃) ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩)))
112	111	rabbidva 3420	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑄⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃) ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩)})
113		simp1 1152	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑁 ∈ ℕ)
114		simp21 1223	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑃 ∈ (𝔼‘𝑁))
115		simp22 1224	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑄 ∈ (𝔼‘𝑁))
116		simp3l 1218	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑃 ≠ 𝑄)
117		fvline2 36592	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → (𝑃Line𝑄) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑄⟩})
118	113, 114, 115, 116, 117	syl13anc 1397	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → (𝑃Line𝑄) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑄⟩})
119	118	adantr 485	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑃Line𝑄) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝑃, 𝑄⟩})
120		fvray 36587	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → (𝑃Ray𝑄) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑄, 𝑥⟩})
121	113, 114, 115, 116, 120	syl13anc 1397	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → (𝑃Ray𝑄) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑄, 𝑥⟩})
122		rabsn 4686	. . . . . . . . 9 ⊢ (𝑃 ∈ (𝔼‘𝑁) → {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 = 𝑃} = {𝑃})
123	114, 122	syl 18	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 = 𝑃} = {𝑃})
124	123	eqcomd 2767	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → {𝑃} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 = 𝑃})
125	121, 124	uneq12d 4122	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → ((𝑃Ray𝑄) ∪ {𝑃}) = ({𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑄, 𝑥⟩} ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 = 𝑃}))
126		simp23 1225	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑅 ∈ (𝔼‘𝑁))
127		simp3r 1219	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑃 ≠ 𝑅)
128		fvray 36587	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑅)) → (𝑃Ray𝑅) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑅, 𝑥⟩})
129	113, 114, 126, 127, 128	syl13anc 1397	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → (𝑃Ray𝑅) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑅, 𝑥⟩})
130	125, 129	uneq12d 4122	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → (((𝑃Ray𝑄) ∪ {𝑃}) ∪ (𝑃Ray𝑅)) = (({𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑄, 𝑥⟩} ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 = 𝑃}) ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑅, 𝑥⟩}))
131	130	adantr 485	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (((𝑃Ray𝑄) ∪ {𝑃}) ∪ (𝑃Ray𝑅)) = (({𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑄, 𝑥⟩} ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 = 𝑃}) ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑅, 𝑥⟩}))
132		unrab 4267	. . . . . 6 ⊢ ({𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑄, 𝑥⟩} ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 = 𝑃}) = {𝑥 ∈ (𝔼‘𝑁) ∣ (𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃)}
133	132	uneq1i 4117	. . . . 5 ⊢ (({𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑄, 𝑥⟩} ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 = 𝑃}) ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑅, 𝑥⟩}) = ({𝑥 ∈ (𝔼‘𝑁) ∣ (𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃)} ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑅, 𝑥⟩})
134		unrab 4267	. . . . 5 ⊢ ({𝑥 ∈ (𝔼‘𝑁) ∣ (𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃)} ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑅, 𝑥⟩}) = {𝑥 ∈ (𝔼‘𝑁) ∣ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃) ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩)}
135	133, 134	eqtri 2784	. . . 4 ⊢ (({𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑄, 𝑥⟩} ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 = 𝑃}) ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑅, 𝑥⟩}) = {𝑥 ∈ (𝔼‘𝑁) ∣ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃) ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩)}
136	131, 135	eqtrdi 2812	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (((𝑃Ray𝑄) ∪ {𝑃}) ∪ (𝑃Ray𝑅)) = {𝑥 ∈ (𝔼‘𝑁) ∣ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃) ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩)})
137	112, 119, 136	3eqtr4d 2806	. 2 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑃 Btwn ⟨𝑄, 𝑅⟩) → (𝑃Line𝑄) = (((𝑃Ray𝑄) ∪ {𝑃}) ∪ (𝑃Ray𝑅)))
138	137	ex 417	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → (𝑃 Btwn ⟨𝑄, 𝑅⟩ → (𝑃Line𝑄) = (((𝑃Ray𝑄) ∪ {𝑃}) ∪ (𝑃Ray𝑅))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∨ w3o 1100 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 ≠ wne 2956 {crab 3414 ∪ cun 3902 {csn 4588 ⟨cop 4594 class class class wbr 5108 ‘cfv 6536 (class class class)co 7410 ℕcn 12232 𝔼cee 29203 Btwn cbtwn 29204 Colinear ccolin 36483 OutsideOfcoutsideof 36565 Linecline2 36580 Raycray 36581

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 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 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 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 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-ec 8695 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9401 df-oi 9471 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-n0 12504 df-z 12591 df-uz 12862 df-rp 13016 df-ico 13377 df-icc 13378 df-fz 13535 df-fzo 13682 df-seq 14037 df-exp 14097 df-hash 14366 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-clim 15538 df-sum 15737 df-ee 29206 df-btwn 29207 df-cgr 29208 df-ofs 36429 df-colinear 36485 df-ifs 36486 df-cgr3 36487 df-fs 36488 df-outsideof 36566 df-line2 36583 df-ray 36584

This theorem is referenced by: (None)

W3C validator