Theorem nrgngp 24577

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 2731	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2731	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 24575	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 497	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2111  ‘cfv 6481  AbsValcabv 20723  normcnm 24491  NrmGrpcngp 24492  NrmRingcnrg 24494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-nrg 24500

This theorem is referenced by:  nrgdsdi  24580  nrgdsdir  24581  unitnmn0  24583  nminvr  24584  nmdvr  24585  nrgtgp  24587  subrgnrg  24588  nlmngp2  24595  sranlm  24599  nrginvrcnlem  24606  nrginvrcn  24607  cnzh  33981  rezh  33982  qqhcn  34004  qqhucn  34005  rrhcn  34010  rrhf  34011  rrexttps  34019  rrexthaus  34020