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Mirrors > Home > MPE Home > Th. List > ringrz | Structured version Visualization version Unicode version |
Description: The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) |
Ref | Expression |
---|---|
rngz.b | |
rngz.t | |
rngz.z |
Ref | Expression |
---|---|
ringrz |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringgrp 18552 | . . . . . 6 | |
2 | rngz.b | . . . . . . . 8 | |
3 | rngz.z | . . . . . . . 8 | |
4 | 2, 3 | grpidcl 17450 | . . . . . . 7 |
5 | eqid 2622 | . . . . . . . 8 | |
6 | 2, 5, 3 | grplid 17452 | . . . . . . 7 |
7 | 4, 6 | mpdan 702 | . . . . . 6 |
8 | 1, 7 | syl 17 | . . . . 5 |
9 | 8 | adantr 481 | . . . 4 |
10 | 9 | oveq2d 6666 | . . 3 |
11 | simpr 477 | . . . . 5 | |
12 | 1, 4 | syl 17 | . . . . . 6 |
13 | 12 | adantr 481 | . . . . 5 |
14 | 11, 13, 13 | 3jca 1242 | . . . 4 |
15 | rngz.t | . . . . 5 | |
16 | 2, 5, 15 | ringdi 18566 | . . . 4 |
17 | 14, 16 | syldan 487 | . . 3 |
18 | 1 | adantr 481 | . . . 4 |
19 | 2, 15 | ringcl 18561 | . . . . 5 |
20 | 13, 19 | mpd3an3 1425 | . . . 4 |
21 | 2, 5, 3 | grplid 17452 | . . . . 5 |
22 | 21 | eqcomd 2628 | . . . 4 |
23 | 18, 20, 22 | syl2anc 693 | . . 3 |
24 | 10, 17, 23 | 3eqtr3d 2664 | . 2 |
25 | 2, 5 | grprcan 17455 | . . 3 |
26 | 18, 20, 13, 20, 25 | syl13anc 1328 | . 2 |
27 | 24, 26 | mpbid 222 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 cmulr 15942 c0g 16100 cgrp 17422 crg 18547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-mgp 18490 df-ring 18549 |
This theorem is referenced by: ringsrg 18589 ringinvnz1ne0 18592 ringinvnzdiv 18593 rngnegr 18595 gsummgp0 18608 gsumdixp 18609 dvdsr02 18656 isdrng2 18757 drngmul0or 18768 cntzsubr 18812 isabvd 18820 lmodvs0 18897 rrgeq0 19290 unitrrg 19293 domneq0 19297 psrridm 19404 mpllsslem 19435 mplsubrglem 19439 mplcoe1 19465 mplmon2 19493 evlslem4 19508 coe1tmmul2 19646 cply1mul 19664 ocvlss 20016 frlmphl 20120 uvcresum 20132 mamurid 20248 matsc 20256 dmatmul 20303 dmatscmcl 20309 scmatscmide 20313 mulmarep1el 20378 mdetdiaglem 20404 mdetero 20416 mdetunilem8 20425 mdetunilem9 20426 mdetuni0 20427 maducoeval2 20446 madugsum 20449 smadiadetlem1a 20469 smadiadetglem2 20478 chpdmatlem2 20644 chfacfpmmul0 20667 cayhamlem4 20693 mdegvscale 23835 mdegmullem 23838 coe1mul3 23859 deg1mul3le 23876 ply1divex 23896 ply1rem 23923 fta1blem 23928 kerunit 29823 matunitlindflem1 33405 lfl0f 34356 lfl0sc 34369 lkrlss 34382 lcfrlem33 36864 hdmapinvlem3 37212 hdmapglem7b 37220 cntzsdrg 37772 mgpsumz 42141 domnmsuppn0 42150 rmsuppss 42151 ply1mulgsumlem2 42175 lincresunit2 42267 |
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