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Metamath Proof Explorer

Theorem zsubcld

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

Ref Expression
Hypotheses zred.1 ⊢ φ → A ∈ ℤ
zaddcld.1 ⊢ φ → B ∈ ℤ
Assertion zsubcld ⊢ φ → A − B ∈ ℤ

Proof

Step Hyp Ref Expression
1 zred.1 ⊢ φ → A ∈ ℤ
2 zaddcld.1 ⊢ φ → B ∈ ℤ
3 zsubcl ⊢ A ∈ ℤ ∧ B ∈ ℤ → A − B ∈ ℤ
4 1 2 3 syl2anc ⊢ φ → A − B ∈ ℤ