Theorem nrgngp 24683

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 2737	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2737	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 24681	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 497	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2108  ‘cfv 6561  AbsValcabv 20809  normcnm 24589  NrmGrpcngp 24590  NrmRingcnrg 24592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-nrg 24598

This theorem is referenced by:  nrgdsdi  24686  nrgdsdir  24687  unitnmn0  24689  nminvr  24690  nmdvr  24691  nrgtgp  24693  subrgnrg  24694  nlmngp2  24701  sranlm  24705  nrginvrcnlem  24712  nrginvrcn  24713  cnzh  33969  rezh  33970  qqhcn  33992  qqhucn  33993  rrhcn  33998  rrhf  33999  rrexttps  34007  rrexthaus  34008