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Mirrors > Home > MPE Home > Th. List > isdrng | Structured version Visualization version Unicode version |
Description: The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
isdrng.b |
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isdrng.u |
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isdrng.z |
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Ref | Expression |
---|---|
isdrng |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5892 |
. . . 4
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2 | isdrng.u |
. . . 4
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3 | 1, 2 | syl6eqr 2514 |
. . 3
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4 | fveq2 5892 |
. . . . 5
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5 | isdrng.b |
. . . . 5
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6 | 4, 5 | syl6eqr 2514 |
. . . 4
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7 | fveq2 5892 |
. . . . . 6
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8 | isdrng.z |
. . . . . 6
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9 | 7, 8 | syl6eqr 2514 |
. . . . 5
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10 | 9 | sneqd 3992 |
. . . 4
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11 | 6, 10 | difeq12d 3564 |
. . 3
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12 | 3, 11 | eqeq12d 2477 |
. 2
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13 | df-drng 18032 |
. 2
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14 | 12, 13 | elrab2 3210 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ral 2754 df-rex 2755 df-rab 2758 df-v 3059 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-br 4419 df-iota 5569 df-fv 5613 df-drng 18032 |
This theorem is referenced by: drngunit 18035 drngui 18036 drngring 18037 isdrng2 18040 drngprop 18041 drngid 18044 opprdrng 18054 drngpropd 18057 issubdrg 18088 drngdomn 18582 fidomndrng 18586 istdrg2 21247 cvsunit 22194 cphreccllem 22211 zrhunitpreima 28833 cntzsdrg 36114 |
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