Theorem nrgngp 22794

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 2799	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2799	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 22792	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 492	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2157  ‘cfv 6101  AbsValcabv 19134  normcnm 22709  NrmGrpcngp 22710  NrmRingcnrg 22712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-nrg 22718

This theorem is referenced by:  nrgdsdi  22797  nrgdsdir  22798  unitnmn0  22800  nminvr  22801  nmdvr  22802  nrgtgp  22804  subrgnrg  22805  nlmngp2  22812  sranlm  22816  nrginvrcnlem  22823  nrginvrcn  22824  cnzh  30530  rezh  30531  qqhcn  30551  qqhucn  30552  rrhcn  30557  rrhf  30558  rrexttps  30566  rrexthaus  30567