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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 17122 | . 2 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ SGrp) | |
2 | sgrpmgm 17112 | . 2 ⊢ (𝑀 ∈ SGrp → 𝑀 ∈ Mgm) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Mgmcmgm 17063 SGrpcsgrp 17106 Mndcmnd 17117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-sgrp 17107 df-mnd 17118 |
This theorem is referenced by: mndcl 17124 mndplusf 17132 srg1zr 18352 ringmgm 18380 chfacfpmmulgsum2 20489 cayhamlem1 20490 ofldchr 29145 idomrootle 36792 ismhm0 41595 mhmismgmhm 41596 c0mgm 41699 c0snmgmhm 41704 c0snmhm 41705 |
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