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Metamath Proof Explorer

Theorem odzcl

Description: The order of a group element is an integer. (Contributed by Mario Carneiro, 28-Feb-2014)

		Ref	Expression
	Assertion	odzcl	$⊢ (N \in ℕ \land A \in ℤ \land A \gcd N = 1) \to {od}_{ℤ} (N) (A) \in ℕ$

Step	Hyp	Ref	Expression
1		odzcllem	$⊢ (N \in ℕ \land A \in ℤ \land A \gcd N = 1) \to ({od}_{ℤ} (N) (A) \in ℕ \land N ∥ (A^{{od}_{ℤ} (N) (A)} - 1))$
2	1	simpld	$⊢ (N \in ℕ \land A \in ℤ \land A \gcd N = 1) \to {od}_{ℤ} (N) (A) \in ℕ$