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Theorem isnrg 24521
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 6882 . . . 4 (π‘Ÿ = 𝑅 β†’ (normβ€˜π‘Ÿ) = (normβ€˜π‘…))
2 isnrg.1 . . . 4 𝑁 = (normβ€˜π‘…)
31, 2eqtr4di 2782 . . 3 (π‘Ÿ = 𝑅 β†’ (normβ€˜π‘Ÿ) = 𝑁)
4 fveq2 6882 . . . 4 (π‘Ÿ = 𝑅 β†’ (AbsValβ€˜π‘Ÿ) = (AbsValβ€˜π‘…))
5 isnrg.2 . . . 4 𝐴 = (AbsValβ€˜π‘…)
64, 5eqtr4di 2782 . . 3 (π‘Ÿ = 𝑅 β†’ (AbsValβ€˜π‘Ÿ) = 𝐴)
73, 6eleq12d 2819 . 2 (π‘Ÿ = 𝑅 β†’ ((normβ€˜π‘Ÿ) ∈ (AbsValβ€˜π‘Ÿ) ↔ 𝑁 ∈ 𝐴))
8 df-nrg 24438 . 2 NrmRing = {π‘Ÿ ∈ NrmGrp ∣ (normβ€˜π‘Ÿ) ∈ (AbsValβ€˜π‘Ÿ)}
97, 8elrab2 3679 1 (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  β€˜cfv 6534  AbsValcabv 20655  normcnm 24429  NrmGrpcngp 24430  NrmRingcnrg 24432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-nrg 24438
This theorem is referenced by:  nrgabv  24522  nrgngp  24523  subrgnrg  24534  tngnrg  24535  cnnrg  24641  zhmnrg  33466
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