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Mirrors > Home > ILE Home > Th. List > islring | Unicode version |
Description: The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.) |
Ref | Expression |
---|---|
islring.b |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
islring.a |
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islring.1 |
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islring.u |
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Ref | Expression |
---|---|
islring |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5554 |
. . . 4
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2 | islring.b |
. . . 4
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3 | 1, 2 | eqtr4di 2244 |
. . 3
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4 | fveq2 5554 |
. . . . . . . 8
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5 | islring.a |
. . . . . . . 8
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6 | 4, 5 | eqtr4di 2244 |
. . . . . . 7
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7 | 6 | oveqd 5935 |
. . . . . 6
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8 | fveq2 5554 |
. . . . . . 7
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9 | islring.1 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 8, 9 | eqtr4di 2244 |
. . . . . 6
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11 | 7, 10 | eqeq12d 2208 |
. . . . 5
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12 | fveq2 5554 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | islring.u |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 12, 13 | eqtr4di 2244 |
. . . . . . 7
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15 | 14 | eleq2d 2263 |
. . . . . 6
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16 | 14 | eleq2d 2263 |
. . . . . 6
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17 | 15, 16 | orbi12d 794 |
. . . . 5
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18 | 11, 17 | imbi12d 234 |
. . . 4
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19 | 3, 18 | raleqbidv 2706 |
. . 3
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20 | 3, 19 | raleqbidv 2706 |
. 2
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21 | df-lring 13687 |
. 2
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22 | 20, 21 | elrab2 2919 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-un 3157 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-iota 5215 df-fv 5262 df-ov 5921 df-lring 13687 |
This theorem is referenced by: lringuplu 13692 |
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