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Theorem mndmgm 17123
 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 17122 . 2 (𝑀 ∈ Mnd → 𝑀 ∈ SGrp)
2 sgrpmgm 17112 . 2 (𝑀 ∈ SGrp → 𝑀 ∈ Mgm)
31, 2syl 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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