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| Description: A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| grpmnd | ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 2 | eqid 2736 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 3 | eqid 2736 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 4 | 1, 2, 3 | isgrp 18958 | . 2 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑎 ∈ (Base‘𝐺)∃𝑚 ∈ (Base‘𝐺)(𝑚(+g‘𝐺)𝑎) = (0g‘𝐺))) | 
| 5 | 4 | simplbi 497 | 1 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 0gc0g 17485 Mndcmnd 18748 Grpcgrp 18952 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-grp 18955 | 
| This theorem is referenced by: grpcl 18960 grpass 18961 grpideu 18963 grpmndd 18965 grpplusf 18967 grpplusfo 18968 grpsgrp 18979 dfgrp2 18981 grpidcl 18984 grplid 18986 grprid 18987 dfgrp3 19058 prdsgrpd 19069 prdsinvgd 19070 mulgaddcom 19117 mulginvcom 19118 mulgz 19121 mulgneg2 19127 mulgass 19130 issubg3 19163 grpissubg 19165 0subg 19170 subgacs 19180 0ghm 19249 pwsdiagghm 19263 cntzsubg 19358 oppggrp 19377 symgsubmefmndALT 19422 psgnunilem5 19513 psgnuni 19518 0subgALT 19587 lsmcntzr 19699 pj1ghm 19722 isabl2 19809 cntrabl 19862 dprdfid 20038 dprdfeq0 20043 dprdlub 20047 dmdprdsplitlem 20058 dprddisj2 20060 dpjidcl 20079 pgpfaclem3 20104 simpgnideld 20120 c0ghm 20462 c0snghm 20465 dsmmsubg 21764 frlm0 21775 mdetunilem7 22625 istgp2 24100 cyc3genpm 33173 isarchi3 33195 reofld 33373 lbslsat 33668 dimkerim 33679 fedgmullem2 33682 primrootscoprbij 42104 grpods 42196 pwssplit4 43106 pwslnmlem2 43110 lcoel0 48350 | 
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