Theorem nrgngp 22466

Description: A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)

Assertion

Ref	Expression
nrgngp	|- ( R e. NrmRing -> R e. NrmGrp )

Proof of Theorem nrgngp

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( norm ` R ) = ( norm ` R )
2		eqid 2622	. . 3 |- (AbsVal ` R ) = (AbsVal ` R )
3	1, 2	isnrg 22464	. 2 |- ( R e. NrmRing <-> ( R e. NrmGrp /\ ( norm ` R ) e. (AbsVal ` R ) ) )
4	3	simplbi 476	1 |- ( R e. NrmRing -> R e. NrmGrp )

Syntax hints:   -> wi 4   e. wcel 1990   ` cfv 5888  AbsValcabv 18816   normcnm 22381  NrmGrpcngp 22382  NrmRingcnrg 22384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-nrg 22390

This theorem is referenced by:  nrgdsdi  22469  nrgdsdir  22470  unitnmn0  22472  nminvr  22473  nmdvr  22474  nrgtgp  22476  subrgnrg  22477  nlmngp2  22484  sranlm  22488  nrginvrcnlem  22495  nrginvrcn  22496  cnzh  30014  rezh  30015  qqhcn  30035  qqhucn  30036  rrhcn  30041  rrhf  30042  rrexttps  30050  rrexthaus  30051