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Theorem nrgngp 18570
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 2388 . . 3  |-  ( norm `  R )  =  (
norm `  R )
2 eqid 2388 . . 3  |-  (AbsVal `  R )  =  (AbsVal `  R )
31, 2isnrg 18568 . 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 1717   ` cfv 5395  AbsValcabv 15832   normcnm 18496  NrmGrpcngp 18497  NrmRingcnrg 18499
This theorem is referenced by:  nrgdsdi  18573  nrgdsdir  18574  unitnmn0  18576  nminvr  18577  nmdvr  18578  nrgtgp  18580  subrgnrg  18581  nlmngp2  18588  sranlm  18592  nrginvrcnlem  18598  nrginvrcn  18599  qqhcn  24175  qqhucn  24176  rrhre  24184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-nrg 18505
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