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Theorem grpsgrp 18991
Description: A group is a semigroup. (Contributed by AV, 28-Aug-2021.)
Assertion
Ref Expression
grpsgrp (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)

Proof of Theorem grpsgrp
StepHypRef Expression
1 grpmnd 18971 . 2 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
2 mndsgrp 18766 . 2 (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)
31, 2syl 17 1 (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Smgrpcsgrp 18744  Mndcmnd 18760  Grpcgrp 18964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-mnd 18761  df-grp 18967
This theorem is referenced by:  dfgrp2  18993  dfgrp3  19070  isarchi3  33177
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