Theorem nrgngp 24643

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 24641	. 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 6496  AbsValcabv 20782  normcnm 24557  NrmGrpcngp 24558  NrmRingcnrg 24560

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 2709

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 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6452  df-fv 6504  df-nrg 24566

This theorem is referenced by:  nrgdsdi  24646  nrgdsdir  24647  unitnmn0  24649  nminvr  24650  nmdvr  24651  nrgtgp  24653  subrgnrg  24654  nlmngp2  24661  sranlm  24665  nrginvrcnlem  24672  nrginvrcn  24673  cnzh  34134  rezh  34135  qqhcn  34157  qqhucn  34158  rrhcn  34163  rrhf  34164  rrexttps  34172  rrexthaus  34173