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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 2798 | . . 3 ⊢ (norm‘𝑅) = (norm‘𝑅) | |
2 | eqid 2798 | . . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅) | |
3 | 1, 2 | nrgabv 23267 | . 2 ⊢ (𝑅 ∈ NrmRing → (norm‘𝑅) ∈ (AbsVal‘𝑅)) |
4 | 2 | abvrcl 19585 | . 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 2111 ‘cfv 6324 Ringcrg 19290 AbsValcabv 19580 normcnm 23183 NrmRingcnrg 23186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fv 6332 df-abv 19581 df-nrg 23192 |
This theorem is referenced by: nrgdsdi 23271 nrgdsdir 23272 nmdvr 23276 nrgtgp 23278 rlmnlm 23294 nrgtrg 23296 nrginvrcnlem 23297 nrginvrcn 23298 nrgtdrg 23299 rlmbn 23965 iistmd 31255 zrhnm 31320 cnzh 31321 rezh 31322 |
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