Theorem nrgngp 25008

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

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2763	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 25006	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 502	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ‘cfv 6555  AbsValcabv 21092  normcnm 24922  NrmGrpcngp 24923  NrmRingcnrg 24925

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6509  df-fv 6563  df-nrg 24931

This theorem is used by:  nrgdsdi  25011  nrgdsdir  25012  unitnmn0  25014  nminvr  25015  nmdvr  25016  nrgtgp  25018  subrgnrg  25019  nlmngp2  25026  sranlm  25030  nrginvrcnlem  25037  nrginvrcn  25038  cnzh  34667  rezh  34668  qqhcn  34690  qqhucn  34691  rrhcn  34696  rrhf  34697  rrexttps  34705  rrexthaus  34706