Theorem nrgngp 24623

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 24621	. 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 6502  AbsValcabv 20758  normcnm 24537  NrmGrpcngp 24538  NrmRingcnrg 24540

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 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6458  df-fv 6510  df-nrg 24546

This theorem is referenced by:  nrgdsdi  24626  nrgdsdir  24627  unitnmn0  24629  nminvr  24630  nmdvr  24631  nrgtgp  24633  subrgnrg  24634  nlmngp2  24641  sranlm  24645  nrginvrcnlem  24652  nrginvrcn  24653  cnzh  34152  rezh  34153  qqhcn  34175  qqhucn  34176  rrhcn  34181  rrhf  34182  rrexttps  34190  rrexthaus  34191