Theorem colineardim1 35022

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 35000	. . 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 5154	. 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 1195	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → 𝐴 ∈ 𝑉)
4		opex 5464	. . . 4 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
5		brcnvg 5878	. . . 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 5149	. . . 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 2822	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
9	8	3anbi2d 1442	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ↔ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛))))
10		opeq1 4873	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
11	10	breq2d 5160	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑎 Btwn ⟨𝑏, 𝑐⟩ ↔ 𝑎 Btwn ⟨𝐵, 𝑐⟩))
12		breq1 5151	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑏 Btwn ⟨𝑐, 𝑎⟩ ↔ 𝐵 Btwn ⟨𝑐, 𝑎⟩))
13		opeq2 4874	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 𝐵⟩)
14	13	breq2d 5160	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑐 Btwn ⟨𝑎, 𝑏⟩ ↔ 𝑐 Btwn ⟨𝑎, 𝐵⟩))
15	11, 12, 14	3orbi123d 1436	. . . . . . . . . 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 632	. . . . . . . . 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 3179	. . . . . . . 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 2822	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝑐 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑛)))
19	18	3anbi3d 1443	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ↔ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛))))
20		opeq2 4874	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
21	20	breq2d 5160	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝑎 Btwn ⟨𝐵, 𝑐⟩ ↔ 𝑎 Btwn ⟨𝐵, 𝐶⟩))
22		opeq1 4873	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐶 → ⟨𝑐, 𝑎⟩ = ⟨𝐶, 𝑎⟩)
23	22	breq2d 5160	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝐵 Btwn ⟨𝑐, 𝑎⟩ ↔ 𝐵 Btwn ⟨𝐶, 𝑎⟩))
24		breq1 5151	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝑐 Btwn ⟨𝑎, 𝐵⟩ ↔ 𝐶 Btwn ⟨𝑎, 𝐵⟩))
25	21, 23, 24	3orbi123d 1436	. . . . . . . . . 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 632	. . . . . . . . 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 3179	. . . . . . . 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 2822	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
29	28	3anbi1d 1441	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛))))
30		breq1 5151	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎 Btwn ⟨𝐵, 𝐶⟩ ↔ 𝐴 Btwn ⟨𝐵, 𝐶⟩))
31		opeq2 4874	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨𝐶, 𝑎⟩ = ⟨𝐶, 𝐴⟩)
32	31	breq2d 5160	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝐵 Btwn ⟨𝐶, 𝑎⟩ ↔ 𝐵 Btwn ⟨𝐶, 𝐴⟩))
33		opeq1 4873	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
34	33	breq2d 5160	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝐶 Btwn ⟨𝑎, 𝐵⟩ ↔ 𝐶 Btwn ⟨𝐴, 𝐵⟩))
35	30, 32, 34	3orbi123d 1436	. . . . . . . . . 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 632	. . . . . . . . 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 3179	. . . . . . . 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 7515	. . . . . . 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 483	. . . . 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 484	. . . . . . 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 28161	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑛))) → 𝑁 = 𝑛)
47	46	adantrrl 723	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)))) → 𝑁 = 𝑛)
48		simprrl 780	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)))) → 𝐴 ∈ (𝔼‘𝑛))
49		fveq2 6889	. . . . . . . . . . . 12 ⊢ (𝑁 = 𝑛 → (𝔼‘𝑁) = (𝔼‘𝑛))
50	49	eleq2d 2820	. . . . . . . . . . 11 ⊢ (𝑁 = 𝑛 → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (𝔼‘𝑛)))
51	48, 50	syl5ibrcom 246	. . . . . . . . . 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 422	. . . . . . 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 3154	. . . . 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 239	. . . 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 241	. . 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 239	. 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 241	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ → 𝐴 ∈ (𝔼‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∨ w3o 1087   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  Vcvv 3475  ⟨cop 4634   class class class wbr 5148  ^◡ccnv 5675  ‘cfv 6541  {coprab 7407  ℕcn 12209  𝔼cee 28136   Btwn cbtwn 28137   Colinear ccolin 34998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-z 12556  df-uz 12820  df-fz 13482  df-ee 28139  df-colinear 35000

This theorem is referenced by:  liness  35106