Theorem nrgngp 24573

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 2728	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2728	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 24571	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 497	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2099  ‘cfv 6543  AbsValcabv 20690  normcnm 24479  NrmGrpcngp 24480  NrmRingcnrg 24482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-iota 6495  df-fv 6551  df-nrg 24488

This theorem is referenced by:  nrgdsdi  24576  nrgdsdir  24577  unitnmn0  24579  nminvr  24580  nmdvr  24581  nrgtgp  24583  subrgnrg  24584  nlmngp2  24591  sranlm  24595  nrginvrcnlem  24602  nrginvrcn  24603  cnzh  33566  rezh  33567  qqhcn  33587  qqhucn  33588  rrhcn  33593  rrhf  33594  rrexttps  33602  rrexthaus  33603