This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Any element raised to the power of its order is 1 . (Contributed by Mario Carneiro, 28-Feb-2014)
|
|
Ref |
Expression |
|
Assertion |
odzid |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ ( 𝐴 gcd 𝑁 ) = 1 ) → 𝑁 ∥ ( ( 𝐴 ↑ ( ( odℤ ‘ 𝑁 ) ‘ 𝐴 ) ) − 1 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
odzcllem |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ ( 𝐴 gcd 𝑁 ) = 1 ) → ( ( ( odℤ ‘ 𝑁 ) ‘ 𝐴 ) ∈ ℕ ∧ 𝑁 ∥ ( ( 𝐴 ↑ ( ( odℤ ‘ 𝑁 ) ‘ 𝐴 ) ) − 1 ) ) ) |
| 2 |
1
|
simprd |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ ( 𝐴 gcd 𝑁 ) = 1 ) → 𝑁 ∥ ( ( 𝐴 ↑ ( ( odℤ ‘ 𝑁 ) ‘ 𝐴 ) ) − 1 ) ) |