Theorem nrgngp 24548

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 24546	. 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 6482  AbsValcabv 20693  normcnm 24462  NrmGrpcngp 24463  NrmRingcnrg 24465

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 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-iota 6438  df-fv 6490  df-nrg 24471

This theorem is referenced by:  nrgdsdi  24551  nrgdsdir  24552  unitnmn0  24554  nminvr  24555  nmdvr  24556  nrgtgp  24558  subrgnrg  24559  nlmngp2  24566  sranlm  24570  nrginvrcnlem  24577  nrginvrcn  24578  cnzh  33951  rezh  33952  qqhcn  33974  qqhucn  33975  rrhcn  33980  rrhf  33981  rrexttps  33989  rrexthaus  33990