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

Theorem isnrg 21042
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  e. NrmRing 
<->  ( R  e. NrmGrp  /\  N  e.  A ) )

Proof of Theorem isnrg
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 fveq2 5856 . . . 4  |-  ( r  =  R  ->  ( norm `  r )  =  ( norm `  R
) )
2 isnrg.1 . . . 4  |-  N  =  ( norm `  R
)
31, 2syl6eqr 2502 . . 3  |-  ( r  =  R  ->  ( norm `  r )  =  N )
4 fveq2 5856 . . . 4  |-  ( r  =  R  ->  (AbsVal `  r )  =  (AbsVal `  R ) )
5 isnrg.2 . . . 4  |-  A  =  (AbsVal `  R )
64, 5syl6eqr 2502 . . 3  |-  ( r  =  R  ->  (AbsVal `  r )  =  A )
73, 6eleq12d 2525 . 2  |-  ( r  =  R  ->  (
( norm `  r )  e.  (AbsVal `  r )  <->  N  e.  A ) )
8 df-nrg 20979 . 2  |- NrmRing  =  {
r  e. NrmGrp  |  ( norm `  r )  e.  (AbsVal `  r ) }
97, 8elrab2 3245 1  |-  ( R  e. NrmRing 
<->  ( R  e. NrmGrp  /\  N  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   ` cfv 5578  AbsValcabv 17339   normcnm 20970  NrmGrpcngp 20971  NrmRingcnrg 20973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-nrg 20979
This theorem is referenced by:  nrgabv  21043  nrgngp  21044  subrgnrg  21055  tngnrg  21056  cnnrg  21161  zhmnrg  27821
  Copyright terms: Public domain W3C validator