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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 )