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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 ℤring ∈ CRing

Proof

Step Hyp Ref Expression
1 cncrng ⊢ ℂfld ∈ CRing
2 zsubrg ⊢ ℤ ∈ ( SubRing ‘ ℂfld )
3 df-zring ⊢ ℤring = ( ℂfld ↾s ℤ )
4 3 subrgcrng ⊢ ( ( ℂfld ∈ CRing ∧ ℤ ∈ ( SubRing ‘ ℂfld ) ) → ℤring ∈ CRing )
5 1 2 4 mp2an ⊢ ℤring ∈ CRing