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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 17909 | . 2 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Smgrp) | |
2 | sgrpmgm 17898 | . 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 2111 Mgmcmgm 17842 Smgrpcsgrp 17892 Mndcmnd 17903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-sgrp 17893 df-mnd 17904 |
This theorem is referenced by: mndcl 17911 mndplusf 17921 mndissubm 17964 grpissubg 18291 srg1zr 19272 ringmgm 19301 chfacfpmmulgsum2 21470 cayhamlem1 21471 ofldchr 30938 idomrootle 40139 ismhm0 44425 mhmismgmhm 44426 c0mgm 44533 c0snmgmhm 44538 c0snmhm 44539 |
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