Theorem odzid

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 ) )