MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnrg Structured version   Visualization version   GIF version

Theorem isnrg 24024
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 = (norm‘𝑅)
isnrg.2 𝐴 = (AbsVal‘𝑅)
Assertion
Ref Expression
isnrg (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁𝐴))

Proof of Theorem isnrg
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6842 . . . 4 (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅))
2 isnrg.1 . . . 4 𝑁 = (norm‘𝑅)
31, 2eqtr4di 2794 . . 3 (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁)
4 fveq2 6842 . . . 4 (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅))
5 isnrg.2 . . . 4 𝐴 = (AbsVal‘𝑅)
64, 5eqtr4di 2794 . . 3 (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴)
73, 6eleq12d 2832 . 2 (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁𝐴))
8 df-nrg 23941 . 2 NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)}
97, 8elrab2 3648 1 (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  cfv 6496  AbsValcabv 20275  normcnm 23932  NrmGrpcngp 23933  NrmRingcnrg 23935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-nrg 23941
This theorem is referenced by:  nrgabv  24025  nrgngp  24026  subrgnrg  24037  tngnrg  24038  cnnrg  24144  zhmnrg  32548
  Copyright terms: Public domain W3C validator