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

Theorem isnrg 24697
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 6907 . . . 4 (𝑟 = 𝑅 → (norm‘𝑟) = (norm‘𝑅))
2 isnrg.1 . . . 4 𝑁 = (norm‘𝑅)
31, 2eqtr4di 2793 . . 3 (𝑟 = 𝑅 → (norm‘𝑟) = 𝑁)
4 fveq2 6907 . . . 4 (𝑟 = 𝑅 → (AbsVal‘𝑟) = (AbsVal‘𝑅))
5 isnrg.2 . . . 4 𝐴 = (AbsVal‘𝑅)
64, 5eqtr4di 2793 . . 3 (𝑟 = 𝑅 → (AbsVal‘𝑟) = 𝐴)
73, 6eleq12d 2833 . 2 (𝑟 = 𝑅 → ((norm‘𝑟) ∈ (AbsVal‘𝑟) ↔ 𝑁𝐴))
8 df-nrg 24614 . 2 NrmRing = {𝑟 ∈ NrmGrp ∣ (norm‘𝑟) ∈ (AbsVal‘𝑟)}
97, 8elrab2 3698 1 (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  cfv 6563  AbsValcabv 20826  normcnm 24605  NrmGrpcngp 24606  NrmRingcnrg 24608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-nrg 24614
This theorem is referenced by:  nrgabv  24698  nrgngp  24699  subrgnrg  24710  tngnrg  24711  cnnrg  24817  zhmnrg  33928
  Copyright terms: Public domain W3C validator