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Mirrors > Home > MPE Home > Th. List > grpidcl | Structured version Visualization version GIF version |
Description: The identity element of a group belongs to the group. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
grpidcl.b | ⊢ 𝐵 = (Base‘𝐺) |
grpidcl.o | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
grpidcl | ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 18760 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
2 | grpidcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpidcl.o | . . 3 ⊢ 0 = (0g‘𝐺) | |
4 | 2, 3 | mndidcl 18576 | . 2 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6497 Basecbs 17088 0gc0g 17326 Mndcmnd 18561 Grpcgrp 18753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-riota 7314 df-ov 7361 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 |
This theorem is referenced by: grpbn0 18784 grprcan 18789 grpid 18791 isgrpid2 18792 grprinv 18806 grpidinv 18812 grpinvid 18813 grpidrcan 18817 grpidlcan 18818 grpidssd 18828 grpinvval2 18835 grpsubid1 18837 imasgrp 18868 mulgcl 18898 mulgz 18909 subg0 18939 subg0cl 18941 issubg4 18952 0subgOLD 18959 nmzsubg 18972 eqgid 18987 qusgrp 18990 qus0 18993 ghmid 19019 ghmpreima 19035 ghmf1 19042 gafo 19081 gaid 19084 gass 19086 gaorber 19093 gastacl 19094 lactghmga 19192 cayleylem2 19200 symgsssg 19254 symgfisg 19255 od1 19346 gexdvds 19371 sylow1lem2 19386 sylow3lem1 19414 lsmdisj2 19469 0frgp 19566 odadd1 19631 torsubg 19637 oddvdssubg 19638 0cyg 19675 prmcyg 19676 telgsums 19775 dprdfadd 19804 dprdz 19814 pgpfac1lem3a 19860 ablsimpgprmd 19899 ring0cl 19995 ringlz 20016 ringrz 20017 f1ghm0to0 20181 kerf1ghm 20184 isdrng2 20210 srng0 20333 lmod0vcl 20366 islmhm2 20514 frgpcyg 20996 ip0l 21056 ocvlss 21092 ascl0 21303 psr0cl 21378 mplsubglem 21421 mhp0cl 21552 mhpaddcl 21557 evl1gsumd 21739 grpvlinv 21760 matinvgcell 21800 mat0dim0 21832 mdetdiag 21964 mdetuni0 21986 chpdmatlem2 22204 chp0mat 22211 istgp2 23458 cldsubg 23478 tgpconncompeqg 23479 tgpconncomp 23480 snclseqg 23483 tgphaus 23484 tgpt1 23485 qustgphaus 23490 tgptsmscls 23517 nrmmetd 23946 nmfval2 23963 nmval2 23964 nmf2 23965 ngpds3 23980 nmge0 23989 nmeq0 23990 nminv 23993 nmmtri 23994 nmrtri 23996 nm0 24001 tngnm 24031 idnghm 24123 nmcn 24223 clmvz 24490 nmoleub2lem2 24495 nglmle 24682 mdeg0 25451 dchrinv 26625 dchr1re 26627 dchrpt 26631 dchrsum2 26632 dchrhash 26635 rpvmasumlem 26851 rpvmasum2 26876 dchrisum0re 26877 ogrpinv0lt 31979 ogrpinvlt 31980 isarchi3 32072 archirng 32073 archirngz 32074 archiabllem1b 32077 rmfsupp2 32122 orngsqr 32146 ornglmulle 32147 orngrmulle 32148 ornglmullt 32149 orngrmullt 32150 orngmullt 32151 ofldchr 32156 isarchiofld 32159 qusker 32188 eqg0el 32196 grplsm0l 32232 nsgqus0 32236 nsgmgclem 32237 ghmqusker 32246 fedgmullem1 32381 qqh0 32622 sconnpi1 33890 lfl0f 37577 lkrlss 37603 lshpkrlem1 37618 lkrin 37672 dvhgrp 39616 isdomn4 40670 fsuppind 40808 fsuppssind 40811 mhpind 40812 rnglz 46268 zrrnghm 46301 evl1at0 46558 |
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