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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 2728 | . . 3 ⊢ (norm‘𝑅) = (norm‘𝑅) | |
2 | eqid 2728 | . . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅) | |
3 | 1, 2 | nrgabv 24591 | . 2 ⊢ (𝑅 ∈ NrmRing → (norm‘𝑅) ∈ (AbsVal‘𝑅)) |
4 | 2 | abvrcl 20701 | . 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 2099 ‘cfv 6548 Ringcrg 20173 AbsValcabv 20696 normcnm 24498 NrmRingcnrg 24501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fv 6556 df-abv 20697 df-nrg 24507 |
This theorem is referenced by: nrgdsdi 24595 nrgdsdir 24596 nmdvr 24600 nrgtgp 24602 rlmnlm 24618 nrgtrg 24620 nrginvrcnlem 24621 nrginvrcn 24622 nrgtdrg 24623 rlmbn 25302 iistmd 33503 zrhnm 33570 cnzh 33571 rezh 33572 |
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