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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 17614 | . 2 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ SGrp) | |
2 | sgrpmgm 17604 | . 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 2157 Mgmcmgm 17555 SGrpcsgrp 17598 Mndcmnd 17609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-nul 4983 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-iota 6064 df-fv 6109 df-ov 6881 df-sgrp 17599 df-mnd 17610 |
This theorem is referenced by: mndcl 17616 mndplusf 17624 srg1zr 18845 ringmgm 18873 chfacfpmmulgsum2 20998 cayhamlem1 20999 ofldchr 30330 idomrootle 38558 ismhm0 42604 mhmismgmhm 42605 c0mgm 42708 c0snmgmhm 42713 c0snmhm 42714 |
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