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Mirrors > Home > MPE Home > Th. List > nrgring | Structured version Visualization version GIF version |
Description: A normed ring is a ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nrgring | ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (norm‘𝑅) = (norm‘𝑅) | |
2 | eqid 2821 | . . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅) | |
3 | 1, 2 | nrgabv 23264 | . 2 ⊢ (𝑅 ∈ NrmRing → (norm‘𝑅) ∈ (AbsVal‘𝑅)) |
4 | 2 | abvrcl 19586 | . 2 ⊢ ((norm‘𝑅) ∈ (AbsVal‘𝑅) → 𝑅 ∈ Ring) |
5 | 3, 4 | syl 17 | 1 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ‘cfv 6350 Ringcrg 19291 AbsValcabv 19581 normcnm 23180 NrmRingcnrg 23183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-xp 5556 df-rel 5557 df-cnv 5558 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fv 6358 df-abv 19582 df-nrg 23189 |
This theorem is referenced by: nrgdsdi 23268 nrgdsdir 23269 nmdvr 23273 nrgtgp 23275 rlmnlm 23291 nrgtrg 23293 nrginvrcnlem 23294 nrginvrcn 23295 nrgtdrg 23296 rlmbn 23958 iistmd 31140 zrhnm 31205 cnzh 31206 rezh 31207 |
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