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Theorem grpsgrp 18922
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 18902 . 2 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
2 mndsgrp 18705 . 2 (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)
31, 2syl 17 1 (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Smgrpcsgrp 18683  Mndcmnd 18699  Grpcgrp 18895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-nul 5308
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-iota 6503  df-fv 6559  df-ov 7427  df-mnd 18700  df-grp 18898
This theorem is referenced by:  dfgrp2  18924  dfgrp3  19000  isarchi3  32913
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