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Metamath Proof Explorer

Theorem zmulcld

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

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

Proof

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