Metamath Proof Explorer


Theorem zmulcld

Description: Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses zred.1 ( 𝜑𝐴 ∈ ℤ )
zaddcld.1 ( 𝜑𝐵 ∈ ℤ )
Assertion zmulcld ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℤ )

Proof

Step Hyp Ref Expression
1 zred.1 ( 𝜑𝐴 ∈ ℤ )
2 zaddcld.1 ( 𝜑𝐵 ∈ ℤ )
3 zmulcl ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 · 𝐵 ) ∈ ℤ )
4 1 2 3 syl2anc ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℤ )