Theorem nrgngp 24648

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 2741	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2741	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 24646	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 498	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2121  ‘cfv 6488  AbsValcabv 20783  normcnm 24562  NrmGrpcngp 24563  NrmRingcnrg 24565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-iota 6444  df-fv 6496  df-nrg 24571

This theorem is referenced by:  nrgdsdi  24651  nrgdsdir  24652  unitnmn0  24654  nminvr  24655  nmdvr  24656  nrgtgp  24658  subrgnrg  24659  nlmngp2  24666  sranlm  24670  nrginvrcnlem  24677  nrginvrcn  24678  cnzh  34162  rezh  34163  qqhcn  34185  qqhucn  34186  rrhcn  34191  rrhf  34192  rrexttps  34200  rrexthaus  34201