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Theorem grpsgrp 18978
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 18958 . 2 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
2 mndsgrp 18753 . 2 (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)
31, 2syl 17 1 (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  Smgrpcsgrp 18731  Mndcmnd 18747  Grpcgrp 18951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-mnd 18748  df-grp 18954
This theorem is referenced by:  dfgrp2  18980  dfgrp3  19057  isarchi3  33194
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