Theorem nrgngp 24608

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 2735	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2735	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 24606	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 497	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6491  AbsValcabv 20743  normcnm 24522  NrmGrpcngp 24523  NrmRingcnrg 24525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6447  df-fv 6499  df-nrg 24531

This theorem is referenced by:  nrgdsdi  24611  nrgdsdir  24612  unitnmn0  24614  nminvr  24615  nmdvr  24616  nrgtgp  24618  subrgnrg  24619  nlmngp2  24626  sranlm  24630  nrginvrcnlem  24637  nrginvrcn  24638  cnzh  34104  rezh  34105  qqhcn  34127  qqhucn  34128  rrhcn  34133  rrhf  34134  rrexttps  34142  rrexthaus  34143