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Mirrors > Home > MPE Home > Th. List > mndmgm | Structured version Visualization version GIF version |
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 18766 | . 2 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Smgrp) | |
2 | sgrpmgm 18750 | . 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 2106 Mgmcmgm 18664 Smgrpcsgrp 18744 Mndcmnd 18760 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-sgrp 18745 df-mnd 18761 |
This theorem is referenced by: mndcl 18768 mndplusf 18778 ismhm0 18816 mhmismgmhm 18817 mndissubm 18833 grpmgmd 18992 grpissubg 19177 srg1zr 20233 ringmgm 20262 c0mgm 20476 c0snmgmhm 20479 c0snmhm 20480 psdmplcl 22184 psdadd 22185 chfacfpmmulgsum2 22887 cayhamlem1 22888 idomrootle 26227 fidomncyc 42522 |
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