Theorem nrgngp 23837

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 23835	. 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅)))
4	3	simplbi 498	1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Syntax hints:   → wi 4   ∈ wcel 2110  ‘cfv 6432  AbsValcabv 20087  normcnm 23743  NrmGrpcngp 23744  NrmRingcnrg 23746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-iota 6390  df-fv 6440  df-nrg 23752

This theorem is referenced by:  nrgdsdi  23840  nrgdsdir  23841  unitnmn0  23843  nminvr  23844  nmdvr  23845  nrgtgp  23847  subrgnrg  23848  nlmngp2  23855  sranlm  23859  nrginvrcnlem  23866  nrginvrcn  23867  cnzh  31929  rezh  31930  qqhcn  31950  qqhucn  31951  rrhcn  31956  rrhf  31957  rrexttps  31965  rrexthaus  31966