Theorem nrgngp 23268

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 2798	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2798	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 23266	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 501	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2111  ‘cfv 6324  AbsValcabv 19580  normcnm 23183  NrmGrpcngp 23184  NrmRingcnrg 23186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-nrg 23192

This theorem is referenced by:  nrgdsdi  23271  nrgdsdir  23272  unitnmn0  23274  nminvr  23275  nmdvr  23276  nrgtgp  23278  subrgnrg  23279  nlmngp2  23286  sranlm  23290  nrginvrcnlem  23297  nrginvrcn  23298  cnzh  31321  rezh  31322  qqhcn  31342  qqhucn  31343  rrhcn  31348  rrhf  31349  rrexttps  31357  rrexthaus  31358