lineunray - Mathbox for Scott Fenton

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Theorem lineunray 36165

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 1192	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
2		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (𝔼‘𝑁))
3		simpl21 1252	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ∈ (𝔼‘𝑁))
4		simpl22 1253	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑄 ∈ (𝔼‘𝑁))
5		brcolinear 36077	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 868	. . . . . . . . . . . . . 14 ⊢ (𝑥 Btwn ⟨𝑃, 𝑄⟩ → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))
9	8	orcd 873	. . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ≠ 𝑄)
12	11	necomd 2987	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑄 ≠ 𝑃)
13	12	adantr 480	. . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . 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 1254	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑅 ∈ (𝔼‘𝑁))
18		btwnconn2 36120	. . . . . . . . . . . . . . . . 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 1381	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 874	. . . . . . . . . . . . 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 456	. . . . . . . . . . . 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 36032	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 867	. . . . . . . . . . . . . . 15 ⊢ (𝑄 Btwn ⟨𝑃, 𝑥⟩ → (𝑄 Btwn ⟨𝑃, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑃, 𝑄⟩))
27	26	orcd 873	. . . . . . . . . . . . . 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 253	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 1431	. . . . . . . . . . 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 240	. . . . . . . . . 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 868	. . . . . . . . . 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 36085	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁)) → 𝑃 Colinear ⟨𝑃, 𝑄⟩)
35	1, 3, 4, 34	syl3anc 1373	. . . . . . . . . . . 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 5122	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑃 → (𝑥 Colinear ⟨𝑃, 𝑄⟩ ↔ 𝑃 Colinear ⟨𝑃, 𝑄⟩))
37	35, 36	syl5ibrcom 247	. . . . . . . . . . 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 480	. . . . . . . . . 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 36089	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 36091	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 859	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 1230	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) ∧ 𝑥 ∈ (𝔼‘𝑁)) → 𝑃 ≠ 𝑅)
46	45	adantr 480	. . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . 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 36042	. . . . . . . . . . . . . . . . 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 1381	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 36090	. . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 456	. . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . 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 36033	. . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . 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 36033	. . . . . . . . . . . . . . 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 36039	. . . . . . . . . . . . . 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 36088	. . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 456	. . . . . . . . . . . 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 859	. . . . . . . . . . 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 859	. . . . . . . . . 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 859	. . . . . . . . 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 212	. . . . . . . 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 1021	. . . . . . . . 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 2933	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ 𝑃 ↔ ¬ 𝑥 = 𝑃)
75	74	anbi1i 624	. . . . . . . . . . 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 1009	. . . . . . . . . . 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 277	. . . . . . . . . 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 912	. . . . . . . . 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 275	. . . . . . . 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 287	. . . . . . 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 36140	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 1150	. . . . . . . . . . . 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 1229	. . . . . . . . . . . . . . 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 2987	. . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . 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 456	. . . . . . . . . . . 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 226	. . . . . . . . . . 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 279	. . . . . . . . . 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 36140	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 1150	. . . . . . . . . . . 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 1230	. . . . . . . . . . . . . . 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 2987	. . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . 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 456	. . . . . . . . . . . 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 226	. . . . . . . . . . 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 279	. . . . . . . . . 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 918	. . . . . . . . 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 480	. . . . . . . 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 915	. . . . . . 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 282	. . . . . 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 870	. . . . . . 7 ⊢ ((𝑥 = 𝑃 ∨ (𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩)) ↔ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩) ∨ 𝑥 = 𝑃))
108		or32 925	. . . . . . 7 ⊢ (((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩) ∨ 𝑥 = 𝑃) ↔ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃) ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩))
109	107, 108	bitri 275	. . . . . 6 ⊢ ((𝑥 = 𝑃 ∨ (𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩)) ↔ ((𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃) ∨ 𝑃OutsideOf⟨𝑅, 𝑥⟩))
110	106, 109	bitrdi 287	. . . . 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 652	. . . 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 3422	. . 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 1136	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑁 ∈ ℕ)
114		simp21 1207	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑃 ∈ (𝔼‘𝑁))
115		simp22 1208	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑄 ∈ (𝔼‘𝑁))
116		simp3l 1202	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑃 ≠ 𝑄)
117		fvline2 36164	. . . . 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 1374	. . . 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 480	. . 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 36159	. . . . . . . 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 1374	. . . . . . 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 4697	. . . . . . . . 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 2741	. . . . . . 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 4144	. . . . . 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 1209	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑅 ∈ (𝔼‘𝑁))
127		simp3r 1203	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → 𝑃 ≠ 𝑅)
128		fvray 36159	. . . . . . 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 1374	. . . . . 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 4144	. . . . 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 480	. . . 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 4290	. . . . . 6 ⊢ ({𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝑄, 𝑥⟩} ∪ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 = 𝑃}) = {𝑥 ∈ (𝔼‘𝑁) ∣ (𝑃OutsideOf⟨𝑄, 𝑥⟩ ∨ 𝑥 = 𝑃)}
133	132	uneq1i 4139	. . . . 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 4290	. . . . 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 2758	. . . 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 2786	. . 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 2780	. 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 412	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 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 {crab 3415 ∪ cun 3924 {csn 4601 ⟨cop 4607 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 ℕcn 12240 𝔼cee 28867 Btwn cbtwn 28868 Colinear ccolin 36055 OutsideOfcoutsideof 36137 Linecline2 36152 Raycray 36153

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-ec 8721 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-n0 12502 df-z 12589 df-uz 12853 df-rp 13009 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-seq 14020 df-exp 14080 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-clim 15504 df-sum 15703 df-ee 28870 df-btwn 28871 df-cgr 28872 df-ofs 36001 df-colinear 36057 df-ifs 36058 df-cgr3 36059 df-fs 36060 df-outsideof 36138 df-line2 36155 df-ray 36156

This theorem is referenced by: (None)

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