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| Description: A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| nrgngp | ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (norm‘𝑅) = (norm‘𝑅) | |
| 2 | eqid 2737 | . . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅) | |
| 3 | 1, 2 | isnrg 24681 | . 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅))) | 
| 4 | 3 | simplbi 497 | 1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ‘cfv 6561 AbsValcabv 20809 normcnm 24589 NrmGrpcngp 24590 NrmRingcnrg 24592 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-nrg 24598 | 
| This theorem is referenced by: nrgdsdi 24686 nrgdsdir 24687 unitnmn0 24689 nminvr 24690 nmdvr 24691 nrgtgp 24693 subrgnrg 24694 nlmngp2 24701 sranlm 24705 nrginvrcnlem 24712 nrginvrcn 24713 cnzh 33969 rezh 33970 qqhcn 33992 qqhucn 33993 rrhcn 33998 rrhf 33999 rrexttps 34007 rrexthaus 34008 | 
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