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Theorem isnrg 24681
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 6906 . . . 4 (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅))
2 isnrg.1 . . . 4 𝑁 = (norm‘𝑅)
31, 2eqtr4di 2795 . . 3 (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁)
4 fveq2 6906 . . . 4 (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅))
5 isnrg.2 . . . 4 𝐴 = (AbsVal‘𝑅)
64, 5eqtr4di 2795 . . 3 (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴)
73, 6eleq12d 2835 . 2 (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁𝐴))
8 df-nrg 24598 . 2 NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)}
97, 8elrab2 3695 1 (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  cfv 6561  AbsValcabv 20809  normcnm 24589  NrmGrpcngp 24590  NrmRingcnrg 24592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-nrg 24598
This theorem is referenced by:  nrgabv  24682  nrgngp  24683  subrgnrg  24694  tngnrg  24695  cnnrg  24801  zhmnrg  33966
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