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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 13265 | . 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing) | |
2 | lringnz.1 | . . 3 ⊢ 1 = (1r‘𝑅) | |
3 | lringnz.2 | . . 3 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | nzrnz 13257 | . 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 1353 ∈ wcel 2148 ≠ wne 2347 ‘cfv 5215 0gc0g 12693 1rcur 13073 NzRingcnzr 13254 LRingclring 13262 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-iota 5177 df-fv 5223 df-nzr 13255 df-lring 13263 |
This theorem is referenced by: aprap 13275 |
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