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Definition df-nrg 23195
 Description: A normed ring is a ring with an induced topology and metric such that the metric is translation-invariant and the norm (distance from 0) is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
df-nrg NrmRing = {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)}

Detailed syntax breakdown of Definition df-nrg
StepHypRef Expression
1 cnrg 23189 . 2 class NrmRing
2 vw . . . . . 6 setvar 𝑤
32cv 1537 . . . . 5 class 𝑤
4 cnm 23186 . . . . 5 class norm
53, 4cfv 6343 . . . 4 class (norm‘𝑤)
6 cabv 19587 . . . . 5 class AbsVal
73, 6cfv 6343 . . . 4 class (AbsVal‘𝑤)
85, 7wcel 2115 . . 3 wff (norm‘𝑤) ∈ (AbsVal‘𝑤)
9 cngp 23187 . . 3 class NrmGrp
108, 2, 9crab 3137 . 2 class {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)}
111, 10wceq 1538 1 wff NrmRing = {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)}
 Colors of variables: wff setvar class This definition is referenced by:  isnrg  23269
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