Theorem nrgngp 23271

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 23269	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 500	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6355  AbsValcabv 19587  normcnm 23186  NrmGrpcngp 23187  NrmRingcnrg 23189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363  df-nrg 23195

This theorem is referenced by:  nrgdsdi  23274  nrgdsdir  23275  unitnmn0  23277  nminvr  23278  nmdvr  23279  nrgtgp  23281  subrgnrg  23282  nlmngp2  23289  sranlm  23293  nrginvrcnlem  23300  nrginvrcn  23301  cnzh  31211  rezh  31212  qqhcn  31232  qqhucn  31233  rrhcn  31238  rrhf  31239  rrexttps  31247  rrexthaus  31248