Theorem nrgngp 23732

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 2738	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2738	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 23730	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 497	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2108  ‘cfv 6418  AbsValcabv 19991  normcnm 23638  NrmGrpcngp 23639  NrmRingcnrg 23641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-nrg 23647

This theorem is referenced by:  nrgdsdi  23735  nrgdsdir  23736  unitnmn0  23738  nminvr  23739  nmdvr  23740  nrgtgp  23742  subrgnrg  23743  nlmngp2  23750  sranlm  23754  nrginvrcnlem  23761  nrginvrcn  23762  cnzh  31820  rezh  31821  qqhcn  31841  qqhucn  31842  rrhcn  31847  rrhf  31848  rrexttps  31856  rrexthaus  31857