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Definition df-nrg 22300
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 22294 . 2 class NrmRing
2 vw . . . . . 6 setvar 𝑤
32cv 1479 . . . . 5 class 𝑤
4 cnm 22291 . . . . 5 class norm
53, 4cfv 5847 . . . 4 class (norm‘𝑤)
6 cabv 18737 . . . . 5 class AbsVal
73, 6cfv 5847 . . . 4 class (AbsVal‘𝑤)
85, 7wcel 1987 . . 3 wff (norm‘𝑤) ∈ (AbsVal‘𝑤)
9 cngp 22292 . . 3 class NrmGrp
108, 2, 9crab 2911 . 2 class {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)}
111, 10wceq 1480 1 wff NrmRing = {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)}
Colors of variables: wff setvar class
This definition is referenced by:  isnrg  22374
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