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Theorem isnrg 24576
Description: A normed ring is a ring with a norm that makes it into a normed group, and such that the norm is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
isnrg.1 𝑁 = (normβ€˜π‘…)
isnrg.2 𝐴 = (AbsValβ€˜π‘…)
Assertion
Ref Expression
isnrg (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))

Proof of Theorem isnrg
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 fveq2 6897 . . . 4 (π‘Ÿ = 𝑅 β†’ (normβ€˜π‘Ÿ) = (normβ€˜π‘…))
2 isnrg.1 . . . 4 𝑁 = (normβ€˜π‘…)
31, 2eqtr4di 2786 . . 3 (π‘Ÿ = 𝑅 β†’ (normβ€˜π‘Ÿ) = 𝑁)
4 fveq2 6897 . . . 4 (π‘Ÿ = 𝑅 β†’ (AbsValβ€˜π‘Ÿ) = (AbsValβ€˜π‘…))
5 isnrg.2 . . . 4 𝐴 = (AbsValβ€˜π‘…)
64, 5eqtr4di 2786 . . 3 (π‘Ÿ = 𝑅 β†’ (AbsValβ€˜π‘Ÿ) = 𝐴)
73, 6eleq12d 2823 . 2 (π‘Ÿ = 𝑅 β†’ ((normβ€˜π‘Ÿ) ∈ (AbsValβ€˜π‘Ÿ) ↔ 𝑁 ∈ 𝐴))
8 df-nrg 24493 . 2 NrmRing = {π‘Ÿ ∈ NrmGrp ∣ (normβ€˜π‘Ÿ) ∈ (AbsValβ€˜π‘Ÿ)}
97, 8elrab2 3685 1 (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  β€˜cfv 6548  AbsValcabv 20695  normcnm 24484  NrmGrpcngp 24485  NrmRingcnrg 24487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-nrg 24493
This theorem is referenced by:  nrgabv  24577  nrgngp  24578  subrgnrg  24589  tngnrg  24590  cnnrg  24696  zhmnrg  33568
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