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Mirrors > Home > MPE Home > Th. List > lringring | Structured version Visualization version GIF version |
Description: A local ring is a ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.) |
Ref | Expression |
---|---|
lringring | ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lringnzr 20441 | . 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing) | |
2 | nzrring 20418 | . 2 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Ringcrg 20138 NzRingcnzr 20414 LRingclring 20438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-in 3950 df-ss 3960 df-nzr 20415 df-lring 20439 |
This theorem is referenced by: lringuplu 20444 |
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