Theorem nrgngp 24704

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 2740	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2740	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 24702	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 497	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2108  ‘cfv 6573  AbsValcabv 20831  normcnm 24610  NrmGrpcngp 24611  NrmRingcnrg 24613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-nrg 24619

This theorem is referenced by:  nrgdsdi  24707  nrgdsdir  24708  unitnmn0  24710  nminvr  24711  nmdvr  24712  nrgtgp  24714  subrgnrg  24715  nlmngp2  24722  sranlm  24726  nrginvrcnlem  24733  nrginvrcn  24734  cnzh  33916  rezh  33917  qqhcn  33937  qqhucn  33938  rrhcn  33943  rrhf  33944  rrexttps  33952  rrexthaus  33953