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Mirrors > Home > MPE Home > Th. List > ringlidm | Structured version Visualization version Unicode version |
Description: The unit element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.) |
Ref | Expression |
---|---|
rngidm.b | |
rngidm.t | |
rngidm.u |
Ref | Expression |
---|---|
ringlidm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngidm.b | . . 3 | |
2 | rngidm.t | . . 3 | |
3 | rngidm.u | . . 3 | |
4 | 1, 2, 3 | ringidmlem 18570 | . 2 |
5 | 4 | simpld 475 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cbs 15857 cmulr 15942 cur 18501 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-mgp 18490 df-ur 18502 df-ring 18549 |
This theorem is referenced by: rngo2times 18576 ringidss 18577 ringcom 18579 ring1eq0 18590 ringinvnzdiv 18593 ringnegl 18594 imasring 18619 opprring 18631 dvdsrid 18651 unitmulcl 18664 unitgrp 18667 1rinv 18679 dvreq1 18693 ringinvdv 18694 isdrng2 18757 drngmul0or 18768 isdrngd 18772 subrginv 18796 issubrg2 18800 abv1z 18832 issrngd 18861 sralmod 19187 unitrrg 19293 asclmul1 19339 asclrhm 19342 psrlmod 19401 psrlidm 19403 mplmonmul 19464 evlslem1 19515 coe1pwmul 19649 mulgrhm 19846 mamulid 20247 madetsumid 20267 1mavmul 20354 m1detdiag 20403 mdetralt 20414 mdetunilem7 20424 mdetuni 20428 mdetmul 20429 m2detleib 20437 chfacfpmmulgsum 20669 cpmadugsumlemB 20679 nrginvrcnlem 22495 cphsubrglem 22977 ply1divex 23896 ress1r 29789 dvrcan5 29793 ornglmullt 29807 orng0le1 29812 isarchiofld 29817 madjusmdetlem1 29893 matunitlindflem1 33405 lfl0 34352 lfladd 34353 eqlkr3 34388 lcfrlem1 36831 hdmapinvlem4 37213 hdmapglem5 37214 mon1psubm 37784 lidldomn1 41921 invginvrid 42148 ply1sclrmsm 42171 ldepsprlem 42261 |
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