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Mirrors > Home > MPE Home > Th. List > nrgngp | Structured version Visualization version GIF version |
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 2610 | . . 3 ⊢ (norm‘𝑅) = (norm‘𝑅) | |
2 | eqid 2610 | . . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅) | |
3 | 1, 2 | isnrg 22274 | . 2 ⊢ (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ (norm‘𝑅) ∈ (AbsVal‘𝑅))) |
4 | 3 | simplbi 475 | 1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ‘cfv 5804 AbsValcabv 18639 normcnm 22191 NrmGrpcngp 22192 NrmRingcnrg 22194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-nrg 22200 |
This theorem is referenced by: nrgdsdi 22279 nrgdsdir 22280 unitnmn0 22282 nminvr 22283 nmdvr 22284 nrgtgp 22286 subrgnrg 22287 nlmngp2 22294 sranlm 22298 nrginvrcnlem 22305 nrginvrcn 22306 cnzh 29342 rezh 29343 qqhcn 29363 qqhucn 29364 rrhcn 29369 rrhf 29370 rrexttps 29378 rrexthaus 29379 |
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