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Mirrors > Home > MPE Home > Th. List > isnirred | Structured version Unicode version |
Description: The property of being a non-irreducible (reducible) element in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
irred.1 |
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irred.2 |
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irred.3 |
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irred.4 |
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irred.5 |
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Ref | Expression |
---|---|
isnirred |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | irred.4 |
. . . . . . 7
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2 | 1 | eleq2i 2532 |
. . . . . 6
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3 | eldif 3447 |
. . . . . 6
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4 | 2, 3 | bitri 249 |
. . . . 5
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5 | 4 | baibr 897 |
. . . 4
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6 | df-ne 2650 |
. . . . . . . . 9
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7 | 6 | ralbii 2839 |
. . . . . . . 8
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8 | ralnex 2852 |
. . . . . . . 8
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9 | 7, 8 | bitri 249 |
. . . . . . 7
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10 | 9 | ralbii 2839 |
. . . . . 6
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11 | ralnex 2852 |
. . . . . 6
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12 | 10, 11 | bitr2i 250 |
. . . . 5
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13 | 12 | a1i 11 |
. . . 4
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14 | 5, 13 | anbi12d 710 |
. . 3
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15 | ioran 490 |
. . 3
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16 | irred.1 |
. . . 4
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17 | irred.2 |
. . . 4
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18 | irred.3 |
. . . 4
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19 | irred.5 |
. . . 4
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20 | 16, 17, 18, 1, 19 | isirred 16915 |
. . 3
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21 | 14, 15, 20 | 3bitr4g 288 |
. 2
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22 | 21 | con1bid 330 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4522 ax-nul 4530 ax-pow 4579 ax-pr 4640 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3397 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-nul 3747 df-if 3901 df-sn 3987 df-pr 3989 df-op 3993 df-uni 4201 df-br 4402 df-opab 4460 df-mpt 4461 df-id 4745 df-xp 4955 df-rel 4956 df-cnv 4957 df-co 4958 df-dm 4959 df-iota 5490 df-fun 5529 df-fv 5535 df-ov 6204 df-irred 16859 |
This theorem is referenced by: irredn0 16919 irredrmul 16923 |
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