Theorem nrgngp 24699

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 2735	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2735	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 24697	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 497	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2106  ‘cfv 6563  AbsValcabv 20826  normcnm 24605  NrmGrpcngp 24606  NrmRingcnrg 24608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-nrg 24614

This theorem is referenced by:  nrgdsdi  24702  nrgdsdir  24703  unitnmn0  24705  nminvr  24706  nmdvr  24707  nrgtgp  24709  subrgnrg  24710  nlmngp2  24717  sranlm  24721  nrginvrcnlem  24728  nrginvrcn  24729  cnzh  33931  rezh  33932  qqhcn  33954  qqhucn  33955  rrhcn  33960  rrhf  33961  rrexttps  33969  rrexthaus  33970