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Theorem mndmgm 17615
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.)
Assertion
Ref Expression
mndmgm (𝑀 ∈ Mnd → 𝑀 ∈ Mgm)

Proof of Theorem mndmgm
StepHypRef Expression
1 mndsgrp 17614 . 2 (𝑀 ∈ Mnd → 𝑀 ∈ SGrp)
2 sgrpmgm 17604 . 2 (𝑀 ∈ SGrp → 𝑀 ∈ Mgm)
31, 2syl 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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