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Metamath Proof Explorer

Theorem zringnrg

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

Ref Expression
Assertion zringnrg ⊢ ℤ ring ∈ NrmRing

Proof

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