Theorem nrgngp 24178

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 2732	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2732	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 24176	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 498	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2106  ‘cfv 6543  AbsValcabv 20423  normcnm 24084  NrmGrpcngp 24085  NrmRingcnrg 24087

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-nrg 24093

This theorem is referenced by:  nrgdsdi  24181  nrgdsdir  24182  unitnmn0  24184  nminvr  24185  nmdvr  24186  nrgtgp  24188  subrgnrg  24189  nlmngp2  24196  sranlm  24200  nrginvrcnlem  24207  nrginvrcn  24208  cnzh  32945  rezh  32946  qqhcn  32966  qqhucn  32967  rrhcn  32972  rrhf  32973  rrexttps  32981  rrexthaus  32982