Theorem colineardim1 36628

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 36606	. . 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 5113	. 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 1213	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → 𝐴 ∈ 𝑉)
4		opex 5443	. . . 4 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
5		brcnvg 5863	. . . 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 598	. . 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 5108	. . . 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 2850	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
9	8	3anbi2d 1469	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ↔ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛))))
10		opeq1 4836	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
11	10	breq2d 5119	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑎 Btwn ⟨𝑏, 𝑐⟩ ↔ 𝑎 Btwn ⟨𝐵, 𝑐⟩))
12		breq1 5110	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑏 Btwn ⟨𝑐, 𝑎⟩ ↔ 𝐵 Btwn ⟨𝑐, 𝑎⟩))
13		opeq2 4837	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 𝐵⟩)
14	13	breq2d 5119	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑐 Btwn ⟨𝑎, 𝑏⟩ ↔ 𝑐 Btwn ⟨𝑎, 𝐵⟩))
15	11, 12, 14	3orbi123d 1463	. . . . . . . . . 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 644	. . . . . . . . 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 3188	. . . . . . . 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 2850	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝑐 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑛)))
19	18	3anbi3d 1470	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ↔ (𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛))))
20		opeq2 4837	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
21	20	breq2d 5119	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝑎 Btwn ⟨𝐵, 𝑐⟩ ↔ 𝑎 Btwn ⟨𝐵, 𝐶⟩))
22		opeq1 4836	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐶 → ⟨𝑐, 𝑎⟩ = ⟨𝐶, 𝑎⟩)
23	22	breq2d 5119	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝐵 Btwn ⟨𝑐, 𝑎⟩ ↔ 𝐵 Btwn ⟨𝐶, 𝑎⟩))
24		breq1 5110	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝑐 Btwn ⟨𝑎, 𝐵⟩ ↔ 𝐶 Btwn ⟨𝑎, 𝐵⟩))
25	21, 23, 24	3orbi123d 1463	. . . . . . . . . 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 644	. . . . . . . . 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 3188	. . . . . . . 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 2850	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
29	28	3anbi1d 1468	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛))))
30		breq1 5110	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎 Btwn ⟨𝐵, 𝐶⟩ ↔ 𝐴 Btwn ⟨𝐵, 𝐶⟩))
31		opeq2 4837	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨𝐶, 𝑎⟩ = ⟨𝐶, 𝐴⟩)
32	31	breq2d 5119	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝐵 Btwn ⟨𝐶, 𝑎⟩ ↔ 𝐵 Btwn ⟨𝐶, 𝐴⟩))
33		opeq1 4836	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
34	33	breq2d 5119	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝐶 Btwn ⟨𝑎, 𝐵⟩ ↔ 𝐶 Btwn ⟨𝐴, 𝐵⟩))
35	30, 32, 34	3orbi123d 1463	. . . . . . . . . 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 644	. . . . . . . . 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 3188	. . . . . . . 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 7526	. . . . . . 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 1143	. . . . . 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 487	. . . . 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 488	. . . . . . 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 1155	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊) → 𝐵 ∈ (𝔼‘𝑁))
43	42	anim2i 629	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)))
44		3simpa 1166	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛)) → (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)))
45	44	anim2i 629	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛))) → (𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛))))
46		axdimuniq 29356	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑛))) → 𝑁 = 𝑛)
47	46	adantrrl 737	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)))) → 𝑁 = 𝑛)
48		simprrl 793	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)))) → 𝐴 ∈ (𝔼‘𝑛))
49		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑁 = 𝑛 → (𝔼‘𝑁) = (𝔼‘𝑛))
50	49	eleq2d 2848	. . . . . . . . . . 11 ⊢ (𝑁 = 𝑛 → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (𝔼‘𝑛)))
51	48, 50	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)))) → (𝑁 = 𝑛 → 𝐴 ∈ (𝔼‘𝑁)))
52	47, 51	mpd 16	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝑛 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛)))) → 𝐴 ∈ (𝔼‘𝑁))
53	43, 45, 52	syl2an 608	. . . . . . . 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 426	. . . . . . 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 75	. . . . . 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 3163	. . . . 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 243	. . . 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 245	. . 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 243	. 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 245	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ 𝑊)) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ → 𝐴 ∈ (𝔼‘𝑁)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  Vcvv 3453  ⟨cop 4593   class class class wbr 5107  ^◡ccnv 5658  ‘cfv 6537  {coprab 7417  ℕcn 12260  𝔼cee 29330   Btwn cbtwn 29331   Colinear ccolin 36604

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-z 12619  df-uz 12891  df-fz 13564  df-ee 29333  df-colinear 36606

This theorem is used by:  liness  36712