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Mirrors > Home > ILE Home > Th. List > ringsrg | Unicode version |
Description: Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
ringsrg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringcmn 13147 |
. 2
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2 | eqid 2177 |
. . 3
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3 | 2 | ringmgp 13116 |
. 2
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4 | eqid 2177 |
. . . . 5
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5 | eqid 2177 |
. . . . 5
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6 | eqid 2177 |
. . . . 5
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7 | 4, 2, 5, 6 | isring 13114 |
. . . 4
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8 | 7 | simp3bi 1014 |
. . 3
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9 | eqid 2177 |
. . . . . 6
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10 | 4, 6, 9 | ringlz 13153 |
. . . . 5
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11 | 4, 6, 9 | ringrz 13154 |
. . . . 5
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12 | 10, 11 | jca 306 |
. . . 4
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13 | 12 | ralrimiva 2550 |
. . 3
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14 | r19.26 2603 |
. . 3
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15 | 8, 13, 14 | sylanbrc 417 |
. 2
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16 | 4, 2, 5, 6, 9 | issrg 13079 |
. 2
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17 | 1, 3, 15, 16 | syl3anbrc 1181 |
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-coll 4117 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-pre-ltirr 7920 ax-pre-ltadd 7924 |
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-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-ltxr 7993 df-inn 8916 df-2 8974 df-3 8975 df-ndx 12457 df-slot 12458 df-base 12460 df-sets 12461 df-plusg 12541 df-mulr 12542 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-grp 12812 df-minusg 12813 df-cmn 13021 df-abl 13022 df-mgp 13062 df-ur 13074 df-srg 13078 df-ring 13112 |
This theorem is referenced by: dvdsrcl2 13199 dvdsrid 13200 dvdsrtr 13201 dvdsrmul1 13202 dvdsrneg 13203 dvdsr01 13204 dvdsr02 13205 1unit 13207 opprunitd 13210 crngunit 13211 unitmulcl 13213 unitmulclb 13214 unitgrp 13216 unitabl 13217 unitgrpid 13218 unitsubm 13219 unitinvcl 13223 unitinvinv 13224 ringinvcl 13225 unitlinv 13226 unitrinv 13227 unitnegcl 13230 dvrvald 13234 unitdvcl 13236 dvrid 13237 dvrcan1 13240 dvrcan3 13241 dvreq1 13242 dvrdir 13243 rdivmuldivd 13244 unitpropdg 13248 invrpropdg 13249 subrgdvds 13294 subrguss 13295 subrginv 13296 subrgunit 13298 subrgugrp 13299 subrgintm 13302 dvdsrzring 13362 |
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