lineunray - Mathbox for Scott Fenton

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Theorem lineunray 32782

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 1246	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
2		simpr 479	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (𝔼‘𝑁))
3		simpl21 1339	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ∈ (𝔼‘𝑁))
4		simpl22 1341	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑄 ∈ (𝔼‘𝑁))
5		brcolinear 32694	. . . . . . . . . . . . 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 1495	. . . . . . . . . . . 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 474	. . . . . . . . . . 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 899	. . . . . . . . . . . . . 14 ⊢ (𝑥 Btwn ⟨𝑃, 𝑄⟩ → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))
9	8	orcd 904	. . . . . . . . . . . . 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 1305	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ≠ 𝑄)
12	11	necomd 3054	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑄 ≠ 𝑃)
13	12	adantr 474	. . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . 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 1162	. . . . . . . . . . . . . . 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 1343	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑅 ∈ (𝔼‘𝑁))
18		btwnconn2 32737	. . . . . . . . . . . . . . . . 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 1502	. . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . 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 15	. . . . . . . . . . . . . 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 905	. . . . . . . . . . . . 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 450	. . . . . . . . . . . 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 32649	. . . . . . . . . . . . . . 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 1495	. . . . . . . . . . . . . 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 898	. . . . . . . . . . . . . . 15 ⊢ (𝑄 Btwn ⟨𝑃, 𝑥⟩ → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))
27	26	orcd 904	. . . . . . . . . . . . . 14 ⊢ (𝑄 Btwn ⟨𝑃, 𝑥⟩ → ((𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩) ∨ (𝑅 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑅⟩)))
28	25, 27	syl6bi 245	. . . . . . . . . . . . 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 474	. . . . . . . . . . . 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 1557	. . . . . . . . . . 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 232	. . . . . . . . . 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 899	. . . . . . . . . 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 35	. . . . . . . . 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 32702	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁)) → 𝑃 Colinear ⟨𝑃, 𝑄⟩)
35	1, 3, 4, 34	syl3anc 1494	. . . . . . . . . . . 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 4876	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑃 → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ 𝑃 Colinear ⟨𝑃, 𝑄⟩))
37	35, 36	syl5ibrcom 239	. . . . . . . . . . 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 474	. . . . . . . . . 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 32706	. . . . . . . . . . . . . 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 1495	. . . . . . . . . . . . 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 32708	. . . . . . . . . . . . . 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 1495	. . . . . . . . . . . . 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 890	. . . . . . . . . . . 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 474	. . . . . . . . . . 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 1307	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ≠ 𝑅)
46	45	adantr 474	. . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . 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 1162	. . . . . . . . . . . . . . 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 32659	. . . . . . . . . . . . . . . . 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 1502	. . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . 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 15	. . . . . . . . . . . . . 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 32707	. . . . . . . . . . . . . . . 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 1495	. . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . 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 15	. . . . . . . . . . . . 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 450	. . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . 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 32650	. . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . 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 32650	. . . . . . . . . . . . . . 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 32656	. . . . . . . . . . . . . 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 32705	. . . . . . . . . . . . . . . 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 1495	. . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . 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 15	. . . . . . . . . . . . 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 450	. . . . . . . . . . . 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 890	. . . . . . . . . . 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 890	. . . . . . . . . 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 890	. . . . . . . . 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 204	. . . . . . . 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 1048	. . . . . . . . 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 3000	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ 𝑃 ↔ ¬ 𝑥 = 𝑃)
75	74	anbi1i 617	. . . . . . . . . . 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 1035	. . . . . . . . . . 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 269	. . . . . . . . . 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 941	. . . . . . . . 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 267	. . . . . . . 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	syl6bb 279	. . . . . . 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 32757	. . . . . . . . . . . 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 1495	. . . . . . . . . . 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 1186	. . . . . . . . . . . 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 1305	. . . . . . . . . . . . . . 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 3054	. . . . . . . . . . . . . 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 799	. . . . . . . . . . . . . 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 800	. . . . . . . . . . . . . 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 1162	. . . . . . . . . . . . 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 450	. . . . . . . . . . . 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 218	. . . . . . . . . . 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 271	. . . . . . . . . 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 32757	. . . . . . . . . . . 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 1495	. . . . . . . . . . 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 1186	. . . . . . . . . . . 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 1307	. . . . . . . . . . . . . . 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 3054	. . . . . . . . . . . . . 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 799	. . . . . . . . . . . . . 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 800	. . . . . . . . . . . . . 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 1162	. . . . . . . . . . . . 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 450	. . . . . . . . . . . 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 218	. . . . . . . . . . 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 271	. . . . . . . . . 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 947	. . . . . . . . 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 474	. . . . . . . 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 944	. . . . . . 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 274	. . . . . 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 901	. . . . . . 7 ⊢ ((𝑥 = 𝑃 ∨ (𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩)) ↔ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩) ∨ 𝑥 = 𝑃))
108		or32 954	. . . . . . 7 ⊢ (((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩) ∨ 𝑥 = 𝑃) ↔ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃) ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩))
109	107, 108	bitri 267	. . . . . 6 ⊢ ((𝑥 = 𝑃 ∨ (𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩)) ↔ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃) ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩))
110	106, 109	syl6bb 279	. . . . 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 642	. . . 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 3401	. . 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 1170	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑁 ∈ ℕ)
114		simp21 1267	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑃 ∈ (𝔼‘𝑁))
115		simp22 1268	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑄 ∈ (𝔼‘𝑁))
116		simp3l 1262	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑃 ≠ 𝑄)
117		fvline2 32781	. . . . 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 1495	. . . 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 474	. . 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 32776	. . . . . . . 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 1495	. . . . . . 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 4474	. . . . . . . . 9 ⊢ (𝑃 ∈ (𝔼‘𝑁) → {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 = 𝑃} = {𝑃})
123	114, 122	syl 17	. . . . . . . 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 2831	. . . . . . 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 3995	. . . . . 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 1269	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑅 ∈ (𝔼‘𝑁))
127		simp3r 1263	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑃 ≠ 𝑅)
128		fvray 32776	. . . . . . 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 1495	. . . . . 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 3995	. . . . 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 474	. . . 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 4127	. . . . . 6 ⊢ ({𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑄, 𝑥⟩} ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 = 𝑃}) = {𝑥 ∈ (𝔼‘𝑁) ∣ (𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃)}
133	132	uneq1i 3990	. . . . 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 4127	. . . . 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 2849	. . . 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	syl6eq 2877	. . 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 2871	. 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 403	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 198 ∧ wa 386 ∨ wo 878 ∨ w3o 1110 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 {crab 3121 ∪ cun 3796 {csn 4397 ⟨cop 4403 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 ℕcn 11350 𝔼cee 26187 Btwn cbtwn 26188 Colinear ccolin 32672 OutsideOfcoutsideof 32754 Linecline2 32769 Raycray 32770

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-ec 8011 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-ico 12469 df-icc 12470 df-fz 12620 df-fzo 12761 df-seq 13096 df-exp 13155 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-sum 14794 df-ee 26190 df-btwn 26191 df-cgr 26192 df-ofs 32618 df-colinear 32674 df-ifs 32675 df-cgr3 32676 df-fs 32677 df-outsideof 32755 df-line2 32772 df-ray 32773

This theorem is referenced by: (None)

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