Theorem nrgngp 24550

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 2729	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2729	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 24548	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 497	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2109  ‘cfv 6511  AbsValcabv 20717  normcnm 24464  NrmGrpcngp 24465  NrmRingcnrg 24467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-nrg 24473

This theorem is referenced by:  nrgdsdi  24553  nrgdsdir  24554  unitnmn0  24556  nminvr  24557  nmdvr  24558  nrgtgp  24560  subrgnrg  24561  nlmngp2  24568  sranlm  24572  nrginvrcnlem  24579  nrginvrcn  24580  cnzh  33958  rezh  33959  qqhcn  33981  qqhucn  33982  rrhcn  33987  rrhf  33988  rrexttps  33996  rrexthaus  33997