Theorem nndivdvdsd

Description: A positive integer divides a natural number if and only if the quotient is a positive integer, a deduction version of nndivdvds . (Contributed by metakunt, 12-May-2024)

	Ref	Expression
Hypotheses	nndivdvdsd.1	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
	nndivdvdsd.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
Assertion	nndivdvdsd	⊢ ( 𝜑 → ( 𝑀 ∥ 𝑁 ↔ ( 𝑁 / 𝑀 ) ∈ ℕ ) )

Proof

Step	Hyp	Ref	Expression
1		nndivdvdsd.1	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
2		nndivdvdsd.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
3		nndivdvds	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ) → ( 𝑀 ∥ 𝑁 ↔ ( 𝑁 / 𝑀 ) ∈ ℕ ) )
4	2 1 3	syl2anc	⊢ ( 𝜑 → ( 𝑀 ∥ 𝑁 ↔ ( 𝑁 / 𝑀 ) ∈ ℕ ) )