Theorem nrgngp 24981

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 2760	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2760	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 24979	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 502	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  ‘cfv 6538  AbsValcabv 21065  normcnm 24895  NrmGrpcngp 24896  NrmRingcnrg 24898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-nrg 24904

This theorem is used by:  nrgdsdi  24984  nrgdsdir  24985  unitnmn0  24987  nminvr  24988  nmdvr  24989  nrgtgp  24991  subrgnrg  24992  nlmngp2  24999  sranlm  25003  nrginvrcnlem  25010  nrginvrcn  25011  cnzh  34600  rezh  34601  qqhcn  34623  qqhucn  34624  rrhcn  34629  rrhf  34630  rrexttps  34638  rrexthaus  34639