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Mirrors > Home > ILE Home > Th. List > lringring | 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 13673 | . 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing) | |
2 | nzrring 13663 | . 2 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2164 Ringcrg 13476 NzRingcnzr 13659 LRingclring 13670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-rab 2481 df-in 3159 df-ss 3166 df-nzr 13660 df-lring 13671 |
This theorem is referenced by: lringuplu 13676 aprcotr 13765 aprap 13766 |
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