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Mirrors > Home > ILE Home > Th. List > ringrz | Unicode version |
Description: The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) |
Ref | Expression |
---|---|
rngz.b |
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rngz.t |
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rngz.z |
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Ref | Expression |
---|---|
ringrz |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringgrp 13137 |
. . . . . 6
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2 | rngz.b |
. . . . . . 7
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3 | rngz.z |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 2, 3 | grpidcl 12858 |
. . . . . 6
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5 | eqid 2177 |
. . . . . . 7
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6 | 2, 5, 3 | grplid 12860 |
. . . . . 6
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7 | 1, 4, 6 | syl2anc2 412 |
. . . . 5
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8 | 7 | adantr 276 |
. . . 4
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9 | 8 | oveq2d 5890 |
. . 3
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10 | simpr 110 |
. . . . 5
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11 | 1, 4 | syl 14 |
. . . . . 6
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12 | 11 | adantr 276 |
. . . . 5
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13 | 10, 12, 12 | 3jca 1177 |
. . . 4
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14 | rngz.t |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 2, 5, 14 | ringdi 13154 |
. . . 4
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16 | 13, 15 | syldan 282 |
. . 3
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17 | 1 | adantr 276 |
. . . 4
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18 | 2, 14 | ringcl 13149 |
. . . . 5
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19 | 12, 18 | mpd3an3 1338 |
. . . 4
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20 | 2, 5, 3 | grplid 12860 |
. . . . 5
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21 | 20 | eqcomd 2183 |
. . . 4
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22 | 17, 19, 21 | syl2anc 411 |
. . 3
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23 | 9, 16, 22 | 3eqtr3d 2218 |
. 2
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24 | 2, 5 | grprcan 12864 |
. . 3
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25 | 17, 19, 12, 19, 24 | syl13anc 1240 |
. 2
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26 | 23, 25 | mpbid 147 |
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-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-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-inn 8918 df-2 8976 df-3 8977 df-ndx 12459 df-slot 12460 df-base 12462 df-sets 12463 df-plusg 12543 df-mulr 12544 df-0g 12697 df-mgm 12729 df-sgrp 12762 df-mnd 12772 df-grp 12834 df-mgp 13084 df-ring 13134 |
This theorem is referenced by: ringsrg 13177 ringinvnz1ne0 13179 ringinvnzdiv 13180 rngnegr 13182 dvdsr02 13227 |
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