Theorem nrgngp 24798

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 2761	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2761	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 24796	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 501	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2141  ‘cfv 6536  AbsValcabv 20890  normcnm 24712  NrmGrpcngp 24713  NrmRingcnrg 24715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-nrg 24721

This theorem is referenced by:  nrgdsdi  24801  nrgdsdir  24802  unitnmn0  24804  nminvr  24805  nmdvr  24806  nrgtgp  24808  subrgnrg  24809  nlmngp2  24816  sranlm  24820  nrginvrcnlem  24827  nrginvrcn  24828  cnzh  34324  rezh  34325  qqhcn  34347  qqhucn  34348  rrhcn  34353  rrhf  34354  rrexttps  34362  rrexthaus  34363