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Definition df-line2 31216
Description: Define the Line function. This function generates the line passing through the distinct points 𝑎 and 𝑏. Adapted from definition 6.14 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 25-Oct-2013.)
Assertion
Ref Expression
df-line2 Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )}
Distinct variable group:   𝑎,𝑏,𝑙,𝑛

Detailed syntax breakdown of Definition df-line2
StepHypRef Expression
1 cline2 31213 . 2 class Line
2 va . . . . . . . 8 setvar 𝑎
32cv 1473 . . . . . . 7 class 𝑎
4 vn . . . . . . . . 9 setvar 𝑛
54cv 1473 . . . . . . . 8 class 𝑛
6 cee 25482 . . . . . . . 8 class 𝔼
75, 6cfv 5786 . . . . . . 7 class (𝔼‘𝑛)
83, 7wcel 1975 . . . . . 6 wff 𝑎 ∈ (𝔼‘𝑛)
9 vb . . . . . . . 8 setvar 𝑏
109cv 1473 . . . . . . 7 class 𝑏
1110, 7wcel 1975 . . . . . 6 wff 𝑏 ∈ (𝔼‘𝑛)
123, 10wne 2775 . . . . . 6 wff 𝑎𝑏
138, 11, 12w3a 1030 . . . . 5 wff (𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏)
14 vl . . . . . . 7 setvar 𝑙
1514cv 1473 . . . . . 6 class 𝑙
163, 10cop 4126 . . . . . . 7 class 𝑎, 𝑏
17 ccolin 31116 . . . . . . . 8 class Colinear
1817ccnv 5023 . . . . . . 7 class Colinear
1916, 18cec 7600 . . . . . 6 class [⟨𝑎, 𝑏⟩] Colinear
2015, 19wceq 1474 . . . . 5 wff 𝑙 = [⟨𝑎, 𝑏⟩] Colinear
2113, 20wa 382 . . . 4 wff ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )
22 cn 10863 . . . 4 class
2321, 4, 22wrex 2892 . . 3 wff 𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )
2423, 2, 9, 14coprab 6524 . 2 class {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )}
251, 24wceq 1474 1 wff Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )}
Colors of variables: wff setvar class
This definition is referenced by:  funline  31221  linedegen  31222  fvline  31223  ellines  31231
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