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Definition df-od 18149
Description: Define the order of an element in a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.) (Revised by AV, 5-Oct-2020.)
Assertion
Ref Expression
df-od od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ {𝑛 ∈ ℕ ∣ (𝑛(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
Distinct variable group:   𝑔,𝑖,𝑛,𝑥

Detailed syntax breakdown of Definition df-od
StepHypRef Expression
1 cod 18145 . 2 class od
2 vg . . 3 setvar 𝑔
3 cvv 3391 . . 3 class V
4 vx . . . 4 setvar 𝑥
52cv 1636 . . . . 5 class 𝑔
6 cbs 16068 . . . . 5 class Base
75, 6cfv 6101 . . . 4 class (Base‘𝑔)
8 vi . . . . 5 setvar 𝑖
9 vn . . . . . . . . 9 setvar 𝑛
109cv 1636 . . . . . . . 8 class 𝑛
114cv 1636 . . . . . . . 8 class 𝑥
12 cmg 17745 . . . . . . . . 9 class .g
135, 12cfv 6101 . . . . . . . 8 class (.g𝑔)
1410, 11, 13co 6874 . . . . . . 7 class (𝑛(.g𝑔)𝑥)
15 c0g 16305 . . . . . . . 8 class 0g
165, 15cfv 6101 . . . . . . 7 class (0g𝑔)
1714, 16wceq 1637 . . . . . 6 wff (𝑛(.g𝑔)𝑥) = (0g𝑔)
18 cn 11305 . . . . . 6 class
1917, 9, 18crab 3100 . . . . 5 class {𝑛 ∈ ℕ ∣ (𝑛(.g𝑔)𝑥) = (0g𝑔)}
208cv 1636 . . . . . . 7 class 𝑖
21 c0 4116 . . . . . . 7 class
2220, 21wceq 1637 . . . . . 6 wff 𝑖 = ∅
23 cc0 10221 . . . . . 6 class 0
24 cr 10220 . . . . . . 7 class
25 clt 10359 . . . . . . 7 class <
2620, 24, 25cinf 8586 . . . . . 6 class inf(𝑖, ℝ, < )
2722, 23, 26cif 4279 . . . . 5 class if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))
288, 19, 27csb 3728 . . . 4 class {𝑛 ∈ ℕ ∣ (𝑛(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))
294, 7, 28cmpt 4923 . . 3 class (𝑥 ∈ (Base‘𝑔) ↦ {𝑛 ∈ ℕ ∣ (𝑛(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
302, 3, 29cmpt 4923 . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ {𝑛 ∈ ℕ ∣ (𝑛(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
311, 30wceq 1637 1 wff od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ {𝑛 ∈ ℕ ∣ (𝑛(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
Colors of variables: wff setvar class
This definition is referenced by:  odfval  18153
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