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Theorem nrgngp 22276
Description: A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
nrgngp (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Proof of Theorem nrgngp
StepHypRef Expression
1 eqid 2610 . . 3 (norm‘𝑅) = (norm‘𝑅)
2 eqid 2610 . . 3 (AbsVal‘𝑅) = (AbsVal‘𝑅)
31, 2isnrg 22274 . 2 (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
43simplbi 475 1 (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  cfv 5804  AbsValcabv 18639  normcnm 22191  NrmGrpcngp 22192  NrmRingcnrg 22194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-nrg 22200
This theorem is referenced by:  nrgdsdi  22279  nrgdsdir  22280  unitnmn0  22282  nminvr  22283  nmdvr  22284  nrgtgp  22286  subrgnrg  22287  nlmngp2  22294  sranlm  22298  nrginvrcnlem  22305  nrginvrcn  22306  cnzh  29342  rezh  29343  qqhcn  29363  qqhucn  29364  rrhcn  29369  rrhf  29370  rrexttps  29378  rrexthaus  29379
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