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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 18693 | . 2 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Smgrp) | |
2 | sgrpmgm 18677 | . 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 2099 Mgmcmgm 18591 Smgrpcsgrp 18671 Mndcmnd 18687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-nul 5300 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-iota 6494 df-fv 6550 df-ov 7417 df-sgrp 18672 df-mnd 18688 |
This theorem is referenced by: mndcl 18695 mndplusf 18705 ismhm0 18740 mhmismgmhm 18741 mndissubm 18752 grpmgmd 18911 grpissubg 19094 srg1zr 20148 ringmgm 20177 c0mgm 20391 c0snmgmhm 20394 c0snmhm 20395 psdmplcl 22079 psdadd 22080 chfacfpmmulgsum2 22760 cayhamlem1 22761 idomrootle 26100 |
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