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Theorem nrgngp 18189
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 2296 . . 3  |-  ( norm `  R )  =  (
norm `  R )
2 eqid 2296 . . 3  |-  (AbsVal `  R )  =  (AbsVal `  R )
31, 2isnrg 18187 . 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 1696   ` cfv 5271  AbsValcabv 15597   normcnm 18115  NrmGrpcngp 18116  NrmRingcnrg 18118
This theorem is referenced by:  nrgdsdi  18192  nrgdsdir  18193  unitnmn0  18195  nminvr  18196  nmdvr  18197  nrgtgp  18199  subrgnrg  18200  nlmngp2  18207  sranlm  18211  nrginvrcnlem  18217  nrginvrcn  18218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-nrg 18124
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