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Metamath Proof Explorer

Theorem znegcld

Description: Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypothesis zred.1 ⊢ ( 𝜑 → 𝐴 ∈ ℤ )
Assertion znegcld ( 𝜑 → - 𝐴 ∈ ℤ )

Proof

Step Hyp Ref Expression
1 zred.1 ⊢ ( 𝜑 → 𝐴 ∈ ℤ )
2 znegcl ⊢ ( 𝐴 ∈ ℤ → - 𝐴 ∈ ℤ )
3 1 2 syl ⊢ ( 𝜑 → - 𝐴 ∈ ℤ )