Theorem colineardim1 36265

Description: If 𝐴 is colinear with 𝐵 and 𝐶, then 𝐴 is in the same space as 𝐵. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
colineardim1	⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ → 𝐴 ∈ (𝔼‘𝑁)))

Proof of Theorem colineardim1

Dummy variables 𝑎 𝑏 𝑐 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-colinear 36243	. . 3 ⊢ Colinear = ◡{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))}
2	1	breqi 5092	. 2 ⊢ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ↔ 𝐴◡{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))}⟨𝐵, 𝐶⟩)
3		simpr1 1196	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → 𝐴 ∈ 𝑉)
4		opex 5415	. . . 4 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
5		brcnvg 5832	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴◡{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))}⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))}𝐴))
6	3, 4, 5	sylancl 587	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (𝐴◡{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))}⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))}𝐴))
7		df-br 5087	. . . 4 ⊢ (⟨𝐵, 𝐶⟩{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))}𝐴 ↔ ⟨⟨𝐵, 𝐶⟩, 𝐴⟩ ∈ {⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))})
8		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
9	8	3anbi2d 1444	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ↔ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛))))
10		opeq1 4817	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
11	10	breq2d 5098	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑎 Btwn ⟨𝑏, 𝑐⟩ ↔ 𝑎 Btwn ⟨𝐵, 𝑐⟩))
12		breq1 5089	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑏 Btwn ⟨𝑐, 𝑎⟩ ↔ 𝐵 Btwn ⟨𝑐, 𝑎⟩))
13		opeq2 4818	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 𝐵⟩)
14	13	breq2d 5098	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑐 Btwn ⟨𝑎, 𝑏⟩ ↔ 𝑐 Btwn ⟨𝑎, 𝐵⟩))
15	11, 12, 14	3orbi123d 1438	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ((𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩) ↔ (𝑎 Btwn ⟨𝐵, 𝑐⟩ ∨ 𝐵 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝐵⟩)))
16	9, 15	anbi12d 633	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩)) ↔ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝐵, 𝑐⟩ ∨ 𝐵 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝐵⟩))))
17	16	rexbidv 3162	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩)) ↔ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝐵, 𝑐⟩ ∨ 𝐵 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝐵⟩))))
18		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝑐 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑛)))
19	18	3anbi3d 1445	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ↔ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛))))
20		opeq2 4818	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
21	20	breq2d 5098	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝑎 Btwn ⟨𝐵, 𝑐⟩ ↔ 𝑎 Btwn ⟨𝐵, 𝐶⟩))
22		opeq1 4817	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐶 → ⟨𝑐, 𝑎⟩ = ⟨𝐶, 𝑎⟩)
23	22	breq2d 5098	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝐵 Btwn ⟨𝑐, 𝑎⟩ ↔ 𝐵 Btwn ⟨𝐶, 𝑎⟩))
24		breq1 5089	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝑐 Btwn ⟨𝑎, 𝐵⟩ ↔ 𝐶 Btwn ⟨𝑎, 𝐵⟩))
25	21, 23, 24	3orbi123d 1438	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ((𝑎 Btwn ⟨𝐵, 𝑐⟩ ∨ 𝐵 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝐵⟩) ↔ (𝑎 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝑎⟩ ∨ 𝐶 Btwn ⟨𝑎, 𝐵⟩)))
26	19, 25	anbi12d 633	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝐵, 𝑐⟩ ∨ 𝐵 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝐵⟩)) ↔ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝑎⟩ ∨ 𝐶 Btwn ⟨𝑎, 𝐵⟩))))
27	26	rexbidv 3162	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝐵, 𝑐⟩ ∨ 𝐵 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝐵⟩)) ↔ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝑎⟩ ∨ 𝐶 Btwn ⟨𝑎, 𝐵⟩))))
28		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
29	28	3anbi1d 1443	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛))))
30		breq1 5089	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎 Btwn ⟨𝐵, 𝐶⟩ ↔ 𝐴 Btwn ⟨𝐵, 𝐶⟩))
31		opeq2 4818	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨𝐶, 𝑎⟩ = ⟨𝐶, 𝐴⟩)
32	31	breq2d 5098	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝐵 Btwn ⟨𝐶, 𝑎⟩ ↔ 𝐵 Btwn ⟨𝐶, 𝐴⟩))
33		opeq1 4817	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
34	33	breq2d 5098	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝐶 Btwn ⟨𝑎, 𝐵⟩ ↔ 𝐶 Btwn ⟨𝐴, 𝐵⟩))
35	30, 32, 34	3orbi123d 1438	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑎 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝑎⟩ ∨ 𝐶 Btwn ⟨𝑎, 𝐵⟩) ↔ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩)))
36	29, 35	anbi12d 633	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝑎⟩ ∨ 𝐶 Btwn ⟨𝑎, 𝐵⟩)) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ∧ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩))))
37	36	rexbidv 3162	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝑎⟩ ∨ 𝐶 Btwn ⟨𝑎, 𝐵⟩)) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ∧ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩))))
38	17, 27, 37	eloprabg 7474	. . . . . . 7 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (⟨⟨𝐵, 𝐶⟩, 𝐴⟩ ∈ {⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ∧ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩))))
39	38	3comr 1126	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊) → (⟨⟨𝐵, 𝐶⟩, 𝐴⟩ ∈ {⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ∧ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩))))
40	39	adantl 481	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (⟨⟨𝐵, 𝐶⟩, 𝐴⟩ ∈ {⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ∧ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩))))
41		simpl 482	. . . . . . 7 ⊢ (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ∧ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩)) → (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)))
42		simp2 1138	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊) → 𝐵 ∈ (𝔼‘𝑁))
43	42	anim2i 618	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)))
44		3simpa 1149	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) → (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)))
45	44	anim2i 618	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛))) → (𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛))))
46		axdimuniq 29002	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑛))) → 𝑁 = 𝑛)
47	46	adantrrl 725	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)))) → 𝑁 = 𝑛)
48		simprrl 781	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)))) → 𝐴 ∈ (𝔼‘𝑛))
49		fveq2 6838	. . . . . . . . . . . 12 ⊢ (𝑁 = 𝑛 → (𝔼‘𝑁) = (𝔼‘𝑛))
50	49	eleq2d 2823	. . . . . . . . . . 11 ⊢ (𝑁 = 𝑛 → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (𝔼‘𝑛)))
51	48, 50	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)))) → (𝑁 = 𝑛 → 𝐴 ∈ (𝔼‘𝑁)))
52	47, 51	mpd 15	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)))) → 𝐴 ∈ (𝔼‘𝑁))
53	43, 45, 52	syl2an 597	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) ∧ (𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)))) → 𝐴 ∈ (𝔼‘𝑁))
54	53	exp32 420	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (𝑛 ∈ ℕ → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) → 𝐴 ∈ (𝔼‘𝑁))))
55	41, 54	syl7 74	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (𝑛 ∈ ℕ → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ∧ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩)) → 𝐴 ∈ (𝔼‘𝑁))))
56	55	rexlimdv 3137	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ∧ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩)) → 𝐴 ∈ (𝔼‘𝑁)))
57	40, 56	sylbid 240	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (⟨⟨𝐵, 𝐶⟩, 𝐴⟩ ∈ {⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))} → 𝐴 ∈ (𝔼‘𝑁)))
58	7, 57	biimtrid 242	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (⟨𝐵, 𝐶⟩{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))}𝐴 → 𝐴 ∈ (𝔼‘𝑁)))
59	6, 58	sylbid 240	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (𝐴◡{⟨⟨𝑏, 𝑐⟩, 𝑎⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn ⟨𝑏, 𝑐⟩ ∨ 𝑏 Btwn ⟨𝑐, 𝑎⟩ ∨ 𝑐 Btwn ⟨𝑎, 𝑏⟩))}⟨𝐵, 𝐶⟩ → 𝐴 ∈ (𝔼‘𝑁)))
60	2, 59	biimtrid 242	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ → 𝐴 ∈ (𝔼‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1086   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3430  ⟨cop 4574   class class class wbr 5086  ^◡ccnv 5627  ‘cfv 6496  {coprab 7365  ℕcn 12171  𝔼cee 28976   Btwn cbtwn 28977   Colinear ccolin 36241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374  ax-un 7686  ax-cnex 11091  ax-resscn 11092  ax-1cn 11093  ax-icn 11094  ax-addcl 11095  ax-addrcl 11096  ax-mulcl 11097  ax-mulrcl 11098  ax-mulcom 11099  ax-addass 11100  ax-mulass 11101  ax-distr 11102  ax-i2m1 11103  ax-1ne0 11104  ax-1rid 11105  ax-rnegex 11106  ax-rrecex 11107  ax-cnre 11108  ax-pre-lttri 11109  ax-pre-lttrn 11110  ax-pre-ltadd 11111  ax-pre-mulgt0 11112

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5523  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5581  df-we 5583  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7321  df-ov 7367  df-oprab 7368  df-mpo 7369  df-om 7815  df-1st 7939  df-2nd 7940  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11178  df-mnf 11179  df-xr 11180  df-ltxr 11181  df-le 11182  df-sub 11376  df-neg 11377  df-nn 12172  df-z 12522  df-uz 12786  df-fz 13459  df-ee 28979  df-colinear 36243

This theorem is referenced by:  liness  36349