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Definition df-odz 15451
Description: Define the order function on the class of integers mod N. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.)
Assertion
Ref Expression
df-odz od = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}, ℝ, < )))
Distinct variable group:   𝑚,𝑛,𝑥

Detailed syntax breakdown of Definition df-odz
StepHypRef Expression
1 codz 15449 . 2 class od
2 vn . . 3 setvar 𝑛
3 cn 11005 . . 3 class
4 vx . . . 4 setvar 𝑥
54cv 1480 . . . . . . 7 class 𝑥
62cv 1480 . . . . . . 7 class 𝑛
7 cgcd 15197 . . . . . . 7 class gcd
85, 6, 7co 6635 . . . . . 6 class (𝑥 gcd 𝑛)
9 c1 9922 . . . . . 6 class 1
108, 9wceq 1481 . . . . 5 wff (𝑥 gcd 𝑛) = 1
11 cz 11362 . . . . 5 class
1210, 4, 11crab 2913 . . . 4 class {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1}
13 vm . . . . . . . . . 10 setvar 𝑚
1413cv 1480 . . . . . . . . 9 class 𝑚
15 cexp 12843 . . . . . . . . 9 class
165, 14, 15co 6635 . . . . . . . 8 class (𝑥𝑚)
17 cmin 10251 . . . . . . . 8 class
1816, 9, 17co 6635 . . . . . . 7 class ((𝑥𝑚) − 1)
19 cdvds 14964 . . . . . . 7 class
206, 18, 19wbr 4644 . . . . . 6 wff 𝑛 ∥ ((𝑥𝑚) − 1)
2120, 13, 3crab 2913 . . . . 5 class {𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}
22 cr 9920 . . . . 5 class
23 clt 10059 . . . . 5 class <
2421, 22, 23cinf 8332 . . . 4 class inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}, ℝ, < )
254, 12, 24cmpt 4720 . . 3 class (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}, ℝ, < ))
262, 3, 25cmpt 4720 . 2 class (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}, ℝ, < )))
271, 26wceq 1481 1 wff od = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}, ℝ, < )))
Colors of variables: wff setvar class
This definition is referenced by:  odzval  15477
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