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Theorem nrgngp 18688
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 2435 . . 3  |-  ( norm `  R )  =  (
norm `  R )
2 eqid 2435 . . 3  |-  (AbsVal `  R )  =  (AbsVal `  R )
31, 2isnrg 18686 . 2  |-  ( R  e. NrmRing 
<->  ( R  e. NrmGrp  /\  ( norm `  R )  e.  (AbsVal `  R )
) )
43simplbi 447 1  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   ` cfv 5446  AbsValcabv 15894   normcnm 18614  NrmGrpcngp 18615  NrmRingcnrg 18617
This theorem is referenced by:  nrgdsdi  18691  nrgdsdir  18692  unitnmn0  18694  nminvr  18695  nmdvr  18696  nrgtgp  18698  subrgnrg  18699  nlmngp2  18706  sranlm  18710  nrginvrcnlem  18716  nrginvrcn  18717  cnzh  24344  rezh  24345  qqhcn  24365  qqhucn  24366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-nrg 18623
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