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Metamath Proof Explorer

Theorem zringcrng

Description: The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019)

Ref Expression
Assertion zringcrng ⊢ ℤ ring ∈ CRing

Proof

Step Hyp Ref Expression
1 cncrng ⊢ ℂ fld ∈ CRing
2 zsubrg ⊢ ℤ ∈ SubRing ⁡ ℂ fld
3 df-zring ⊢ ℤ ring = ℂ fld ↾ 𝑠 ℤ
4 3 subrgcrng ⊢ ℂ fld ∈ CRing ∧ ℤ ∈ SubRing ⁡ ℂ fld → ℤ ring ∈ CRing
5 1 2 4 mp2an ⊢ ℤ ring ∈ CRing