Theorem nrgngp 23270

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 2821	. . 3 ⊢ (norm‘𝑅) = (norm‘𝑅)
2		eqid 2821	. . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
3	1, 2	isnrg 23268	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 500	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2110  ‘cfv 6354  AbsValcabv 19586  normcnm 23185  NrmGrpcngp 23186  NrmRingcnrg 23188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-iota 6313  df-fv 6362  df-nrg 23194

This theorem is referenced by:  nrgdsdi  23273  nrgdsdir  23274  unitnmn0  23276  nminvr  23277  nmdvr  23278  nrgtgp  23280  subrgnrg  23281  nlmngp2  23288  sranlm  23292  nrginvrcnlem  23299  nrginvrcn  23300  cnzh  31211  rezh  31212  qqhcn  31232  qqhucn  31233  rrhcn  31238  rrhf  31239  rrexttps  31247  rrexthaus  31248