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Metamath Proof Explorer

Theorem nrgabv

Description: The norm of a normed ring is an absolute value. (Contributed by Mario Carneiro, 4-Oct-2015)

Ref Expression
Hypotheses isnrg.1 ⊢ N = norm ⁡ R
isnrg.2 ⊢ A = AbsVal ⁡ R
Assertion nrgabv ⊢ R ∈ NrmRing → N ∈ A

Proof

Step Hyp Ref Expression
1 isnrg.1 ⊢ N = norm ⁡ R
2 isnrg.2 ⊢ A = AbsVal ⁡ R
3 1 2 isnrg ⊢ R ∈ NrmRing ↔ R ∈ NrmGrp ∧ N ∈ A
4 3 simprbi ⊢ R ∈ NrmRing → N ∈ A