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Theorem grpsgrp 18888
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 18868 . 2 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
2 mndsgrp 18671 . 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 18649  Mndcmnd 18665  Grpcgrp 18861
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 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-mnd 18666  df-grp 18864
This theorem is referenced by:  dfgrp2  18890  dfgrp3  18965  isarchi3  32837
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