Theorem nrgngp 24179

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 2733	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2733	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 24177	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 499	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2107  ‘cfv 6544  AbsValcabv 20424  normcnm 24085  NrmGrpcngp 24086  NrmRingcnrg 24088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-nrg 24094

This theorem is referenced by:  nrgdsdi  24182  nrgdsdir  24183  unitnmn0  24185  nminvr  24186  nmdvr  24187  nrgtgp  24189  subrgnrg  24190  nlmngp2  24197  sranlm  24201  nrginvrcnlem  24208  nrginvrcn  24209  cnzh  32950  rezh  32951  qqhcn  32971  qqhucn  32972  rrhcn  32977  rrhf  32978  rrexttps  32986  rrexthaus  32987