Theorem nrgngp 24615

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 24613	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 496	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6487  AbsValcabv 20774  normcnm 24529  NrmGrpcngp 24530  NrmRingcnrg 24532

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 3388  df-v 3429  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-iota 6443  df-fv 6495  df-nrg 24538

This theorem is referenced by:  nrgdsdi  24618  nrgdsdir  24619  unitnmn0  24621  nminvr  24622  nmdvr  24623  nrgtgp  24625  subrgnrg  24626  nlmngp2  24633  sranlm  24637  nrginvrcnlem  24644  nrginvrcn  24645  cnzh  34100  rezh  34101  qqhcn  34123  qqhucn  34124  rrhcn  34129  rrhf  34130  rrexttps  34138  rrexthaus  34139