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

Theorem nrgngp 18173
Description: A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
nrgngp  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )

Proof of Theorem nrgngp
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( norm `  R )  =  (
norm `  R )
2 eqid 2283 . . 3  |-  (AbsVal `  R )  =  (AbsVal `  R )
31, 2isnrg 18171 . 2  |-  ( R  e. NrmRing 
<->  ( R  e. NrmGrp  /\  ( norm `  R )  e.  (AbsVal `  R )
) )
43simplbi 446 1  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   ` cfv 5255  AbsValcabv 15581   normcnm 18099  NrmGrpcngp 18100  NrmRingcnrg 18102
This theorem is referenced by:  nrgdsdi  18176  nrgdsdir  18177  unitnmn0  18179  nminvr  18180  nmdvr  18181  nrgtgp  18183  subrgnrg  18184  nlmngp2  18191  sranlm  18195  nrginvrcnlem  18201  nrginvrcn  18202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-nrg 18108
  Copyright terms: Public domain W3C validator