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Theorem mndmgm 17228
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 17227 . 2 (𝑀 ∈ Mnd → 𝑀 ∈ SGrp)
2 sgrpmgm 17217 . 2 (𝑀 ∈ SGrp → 𝑀 ∈ Mgm)
31, 2syl 17 1 (𝑀 ∈ Mnd → 𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  Mgmcmgm 17168  SGrpcsgrp 17211  Mndcmnd 17222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-ov 6613  df-sgrp 17212  df-mnd 17223
This theorem is referenced by:  mndcl  17229  mndplusf  17237  srg1zr  18457  ringmgm  18485  chfacfpmmulgsum2  20598  cayhamlem1  20599  ofldchr  29617  idomrootle  37281  ismhm0  41114  mhmismgmhm  41115  c0mgm  41218  c0snmgmhm  41223  c0snmhm  41224
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