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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 2610 | . . 3 ⊢ (norm‘𝑅) = (norm‘𝑅) | |
2 | eqid 2610 | . . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅) | |
3 | 1, 2 | nrgabv 22275 | . 2 ⊢ (𝑅 ∈ NrmRing → (norm‘𝑅) ∈ (AbsVal‘𝑅)) |
4 | 2 | abvrcl 18644 | . 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 1977 ‘cfv 5804 Ringcrg 18370 AbsValcabv 18639 normcnm 22191 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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
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-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 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-opab 4644 df-mpt 4645 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fv 5812 df-abv 18640 df-nrg 22200 |
This theorem is referenced by: nrgdsdi 22279 nrgdsdir 22280 nmdvr 22284 nrgtgp 22286 rlmnlm 22302 nrgtrg 22304 nrginvrcnlem 22305 nrginvrcn 22306 nrgtdrg 22307 rlmbn 22965 iistmd 29276 zrhnm 29341 cnzh 29342 rezh 29343 |
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