Theorem gcd2n0cl

Description: Closure of the gcd operator if the second operand is not 0. (Contributed by AV, 10-Jul-2021)

	Ref	Expression
Assertion	gcd2n0cl	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 𝑀 gcd 𝑁 ) ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		neneq	⊢ ( 𝑁 ≠ 0 → ¬ 𝑁 = 0 )
2	1	intnand	⊢ ( 𝑁 ≠ 0 → ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) )
3	2	anim2i	⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 𝑁 ≠ 0 ) → ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) )
4	3	3impa	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) )
5		gcdn0cl	⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) → ( 𝑀 gcd 𝑁 ) ∈ ℕ )
6	4 5	syl	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ) → ( 𝑀 gcd 𝑁 ) ∈ ℕ )