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| Description: A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.) | 
| Ref | Expression | 
|---|---|
| mndmgm | ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mgm) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mndsgrp 18753 | . 2 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Smgrp) | |
| 2 | sgrpmgm 18737 | . 2 ⊢ (𝑀 ∈ Smgrp → 𝑀 ∈ Mgm) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mgm) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 Mgmcmgm 18651 Smgrpcsgrp 18731 Mndcmnd 18747 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-sgrp 18732 df-mnd 18748 | 
| This theorem is referenced by: mndcl 18755 mndplusf 18765 ismhm0 18803 mhmismgmhm 18804 mndissubm 18820 grpmgmd 18979 grpissubg 19164 srg1zr 20212 ringmgm 20241 c0mgm 20459 c0snmgmhm 20462 c0snmhm 20463 psdmplcl 22166 psdadd 22167 psdpw 22174 chfacfpmmulgsum2 22871 cayhamlem1 22872 idomrootle 26212 fidomncyc 42545 | 
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