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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 13673 | . 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing) | |
2 | lringnz.1 | . . 3 ⊢ 1 = (1r‘𝑅) | |
3 | lringnz.2 | . . 3 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | nzrnz 13662 | . 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 1364 ∈ wcel 2164 ≠ wne 2364 ‘cfv 5246 0gc0g 12857 1rcur 13439 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-in1 615 ax-in2 616 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-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-rex 2478 df-rab 2481 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-iota 5207 df-fv 5254 df-nzr 13660 df-lring 13671 |
This theorem is referenced by: aprap 13766 |
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