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Theorem sgrpcl 18739
Description: Closure of the operation of a semigroup. (Contributed by AV, 15-Feb-2025.)
Hypotheses
Ref Expression
sgrpass.b 𝐵 = (Base‘𝐺)
sgrpass.o = (+g𝐺)
Assertion
Ref Expression
sgrpcl ((𝐺 ∈ Smgrp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)

Proof of Theorem sgrpcl
StepHypRef Expression
1 sgrpmgm 18737 . 2 (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
2 sgrpass.b . . 3 𝐵 = (Base‘𝐺)
3 sgrpass.o . . 3 = (+g𝐺)
42, 3mgmcl 18656 . 2 ((𝐺 ∈ Mgm ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
51, 4syl3an1 1164 1 ((𝐺 ∈ Smgrp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Mgmcmgm 18651  Smgrpcsgrp 18731
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-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-mgm 18653  df-sgrp 18732
This theorem is referenced by:  sgrppropd  18744  prdsplusgsgrpcl  18745  cntzsgrpcl  19352  rngpropd  20171
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