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Theorem nrgngp 22406
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 2621 . . 3 (norm‘𝑅) = (norm‘𝑅)
2 eqid 2621 . . 3 (AbsVal‘𝑅) = (AbsVal‘𝑅)
31, 2isnrg 22404 . 2 (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
43simplbi 476 1 (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  cfv 5857  AbsValcabv 18756  normcnm 22321  NrmGrpcngp 22322  NrmRingcnrg 22324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-nrg 22330
This theorem is referenced by:  nrgdsdi  22409  nrgdsdir  22410  unitnmn0  22412  nminvr  22413  nmdvr  22414  nrgtgp  22416  subrgnrg  22417  nlmngp2  22424  sranlm  22428  nrginvrcnlem  22435  nrginvrcn  22436  cnzh  29838  rezh  29839  qqhcn  29859  qqhucn  29860  rrhcn  29865  rrhf  29866  rrexttps  29874  rrexthaus  29875
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