Theorem nrgngp 24399

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 2730	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2730	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 24397	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 496	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2104  ‘cfv 6542  AbsValcabv 20567  normcnm 24305  NrmGrpcngp 24306  NrmRingcnrg 24308

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-nrg 24314

This theorem is referenced by:  nrgdsdi  24402  nrgdsdir  24403  unitnmn0  24405  nminvr  24406  nmdvr  24407  nrgtgp  24409  subrgnrg  24410  nlmngp2  24417  sranlm  24421  nrginvrcnlem  24428  nrginvrcn  24429  cnzh  33248  rezh  33249  qqhcn  33269  qqhucn  33270  rrhcn  33275  rrhf  33276  rrexttps  33284  rrexthaus  33285