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Mirrors > Home > ILE Home > Th. List > lringnz | GIF version |
Description: A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.) |
Ref | Expression |
---|---|
lringnz.1 | ⊢ 1 = (1r‘𝑅) |
lringnz.2 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
lringnz | ⊢ (𝑅 ∈ LRing → 1 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lringnzr 13470 | . 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing) | |
2 | lringnz.1 | . . 3 ⊢ 1 = (1r‘𝑅) | |
3 | lringnz.2 | . . 3 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | nzrnz 13462 | . 2 ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 ) |
5 | 1, 4 | syl 14 | 1 ⊢ (𝑅 ∈ LRing → 1 ≠ 0 ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 ≠ wne 2357 ‘cfv 5228 0gc0g 12723 1rcur 13268 NzRingcnzr 13459 LRingclring 13467 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-rex 2471 df-rab 2474 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-iota 5190 df-fv 5236 df-nzr 13460 df-lring 13468 |
This theorem is referenced by: aprap 13532 |
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