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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 18104 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
2 | grpidcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpidcl.o | . . 3 ⊢ 0 = (0g‘𝐺) | |
4 | 2, 3 | mndidcl 17920 | . 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 1533 ∈ wcel 2110 ‘cfv 6349 Basecbs 16477 0gc0g 16707 Mndcmnd 17905 Grpcgrp 18097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-riota 7108 df-ov 7153 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 |
This theorem is referenced by: grpbn0 18126 grprcan 18131 grpid 18133 isgrpid2 18134 grprinv 18147 grpidinv 18153 grpinvid 18154 grpidrcan 18158 grpidlcan 18159 grpidssd 18169 grpinvval2 18176 grpsubid1 18178 imasgrp 18209 mulgcl 18239 mulgz 18249 subg0 18279 subg0cl 18281 issubg4 18292 0subg 18298 nmzsubg 18311 eqgid 18326 qusgrp 18329 qus0 18332 ghmid 18358 ghmpreima 18374 ghmf1 18381 gafo 18420 gaid 18423 gass 18425 gaorber 18432 gastacl 18433 lactghmga 18527 cayleylem2 18535 symgsssg 18589 symgfisg 18590 od1 18680 gexdvds 18703 sylow1lem2 18718 sylow3lem1 18746 lsmdisj2 18802 0frgp 18899 odadd1 18962 torsubg 18968 oddvdssubg 18969 0cyg 19007 prmcyg 19008 telgsums 19107 dprdfadd 19136 dprdz 19146 pgpfac1lem3a 19192 ablsimpgprmd 19231 ring0cl 19313 ringlz 19331 ringrz 19332 f1ghm0to0 19486 kerf1ghm 19491 kerf1hrmOLD 19492 isdrng2 19506 srng0 19625 lmod0vcl 19657 islmhm2 19804 ascl0 20107 psr0cl 20168 mplsubglem 20208 mhp0cl 20331 mhpaddcl 20332 mhpinvcl 20333 evl1gsumd 20514 frgpcyg 20714 ip0l 20774 ocvlss 20810 grpvlinv 21000 matinvgcell 21038 mat0dim0 21070 mdetdiag 21202 mdetuni0 21224 chpdmatlem2 21441 chp0mat 21448 istgp2 22693 cldsubg 22713 tgpconncompeqg 22714 tgpconncomp 22715 snclseqg 22718 tgphaus 22719 tgpt1 22720 qustgphaus 22725 tgptsmscls 22752 nrmmetd 23178 nmfval2 23194 nmval2 23195 nmf2 23196 ngpds3 23211 nmge0 23220 nmeq0 23221 nminv 23224 nmmtri 23225 nmrtri 23227 nm0 23232 tngnm 23254 idnghm 23346 nmcn 23446 clmvz 23709 nmoleub2lem2 23714 nglmle 23899 mdeg0 24658 dchrinv 25831 dchr1re 25833 dchrpt 25837 dchrsum2 25838 dchrhash 25841 rpvmasumlem 26057 rpvmasum2 26082 dchrisum0re 26083 ogrpinv0lt 30718 ogrpinvlt 30719 isarchi3 30811 archirng 30812 archirngz 30813 archiabllem1b 30816 rmfsupp2 30861 orngsqr 30872 ornglmulle 30873 orngrmulle 30874 ornglmullt 30875 orngrmullt 30876 orngmullt 30877 ofldchr 30882 isarchiofld 30885 qusker 30913 eqg0el 30921 fedgmullem1 31020 qqh0 31220 sconnpi1 32481 lfl0f 36199 lkrlss 36225 lshpkrlem1 36240 lkrin 36294 dvhgrp 38237 rnglz 44149 zrrnghm 44182 evl1at0 44439 |
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