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Definition df-odz 15308
 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 15306 . 2 class od
2 vn . . 3 setvar 𝑛
3 cn 10897 . . 3 class
4 vx . . . 4 setvar 𝑥
54cv 1474 . . . . . . 7 class 𝑥
62cv 1474 . . . . . . 7 class 𝑛
7 cgcd 15054 . . . . . . 7 class gcd
85, 6, 7co 6549 . . . . . 6 class (𝑥 gcd 𝑛)
9 c1 9816 . . . . . 6 class 1
108, 9wceq 1475 . . . . 5 wff (𝑥 gcd 𝑛) = 1
11 cz 11254 . . . . 5 class
1210, 4, 11crab 2900 . . . 4 class {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1}
13 vm . . . . . . . . . 10 setvar 𝑚
1413cv 1474 . . . . . . . . 9 class 𝑚
15 cexp 12722 . . . . . . . . 9 class
165, 14, 15co 6549 . . . . . . . 8 class (𝑥𝑚)
17 cmin 10145 . . . . . . . 8 class
1816, 9, 17co 6549 . . . . . . 7 class ((𝑥𝑚) − 1)
19 cdvds 14821 . . . . . . 7 class
206, 18, 19wbr 4583 . . . . . 6 wff 𝑛 ∥ ((𝑥𝑚) − 1)
2120, 13, 3crab 2900 . . . . 5 class {𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}
22 cr 9814 . . . . 5 class
23 clt 9953 . . . . 5 class <
2421, 22, 23cinf 8230 . . . 4 class inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}, ℝ, < )
254, 12, 24cmpt 4643 . . 3 class (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}, ℝ, < ))
262, 3, 25cmpt 4643 . 2 class (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥𝑚) − 1)}, ℝ, < )))
271, 26wceq 1475 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  15334
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