Theorem nrgngp 24942

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 24940	. 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 2145  ‘cfv 6527  AbsValcabv 21026  normcnm 24856  NrmGrpcngp 24857  NrmRingcnrg 24859

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6483  df-fv 6535  df-nrg 24865

This theorem is used by:  nrgdsdi  24945  nrgdsdir  24946  unitnmn0  24948  nminvr  24949  nmdvr  24950  nrgtgp  24952  subrgnrg  24953  nlmngp2  24960  sranlm  24964  nrginvrcnlem  24971  nrginvrcn  24972  cnzh  34533  rezh  34534  qqhcn  34556  qqhucn  34557  rrhcn  34562  rrhf  34563  rrexttps  34571  rrexthaus  34572