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Mirrors > Home > ILE Home > Th. List > df-lring | Unicode version |
Description: A local ring is a nonzero ring where for any two elements summing to one, at least one is invertible. Any field is a local ring; the ring of integers is an example of a ring which is not a local ring. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.) |
Ref | Expression |
---|---|
df-lring |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clring 13262 |
. 2
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2 | vx |
. . . . . . . . 9
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3 | 2 | cv 1352 |
. . . . . . . 8
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4 | vy |
. . . . . . . . 9
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5 | 4 | cv 1352 |
. . . . . . . 8
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6 | vr |
. . . . . . . . . 10
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7 | 6 | cv 1352 |
. . . . . . . . 9
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8 | cplusg 12528 |
. . . . . . . . 9
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9 | 7, 8 | cfv 5215 |
. . . . . . . 8
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10 | 3, 5, 9 | co 5872 |
. . . . . . 7
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11 | cur 13073 |
. . . . . . . 8
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12 | 7, 11 | cfv 5215 |
. . . . . . 7
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13 | 10, 12 | wceq 1353 |
. . . . . 6
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14 | cui 13187 |
. . . . . . . . 9
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15 | 7, 14 | cfv 5215 |
. . . . . . . 8
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16 | 3, 15 | wcel 2148 |
. . . . . . 7
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17 | 5, 15 | wcel 2148 |
. . . . . . 7
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18 | 16, 17 | wo 708 |
. . . . . 6
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19 | 13, 18 | wi 4 |
. . . . 5
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20 | cbs 12454 |
. . . . . 6
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21 | 7, 20 | cfv 5215 |
. . . . 5
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22 | 19, 4, 21 | wral 2455 |
. . . 4
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23 | 22, 2, 21 | wral 2455 |
. . 3
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24 | cnzr 13254 |
. . 3
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25 | 23, 6, 24 | crab 2459 |
. 2
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26 | 1, 25 | wceq 1353 |
1
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Colors of variables: wff set class |
This definition is referenced by: islring 13264 lringnzr 13265 |
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