Theorem sgrpcl 18663

Description: Closure of the operation of a semigroup. (Contributed by AV, 15-Feb-2025.)

Hypotheses

Ref	Expression
sgrpass.b	⊢ 𝐵 = (Base‘𝐺)
sgrpass.o	⊢ ⚬ = (+g‘𝐺)

Assertion

Ref	Expression
sgrpcl	⊢ ((𝐺 ∈ Smgrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)

Proof of Theorem sgrpcl

Step	Hyp	Ref	Expression
1		sgrpmgm 18661	. 2 ⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
2		sgrpass.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3		sgrpass.o	. . 3 ⊢ ⚬ = (+g‘𝐺)
4	2, 3	mgmcl 18580	. 2 ⊢ ((𝐺 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)
5	1, 4	syl3an1 1164	1 ⊢ ((𝐺 ∈ Smgrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  Mgmcmgm 18575  Smgrpcsgrp 18655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-mgm 18577  df-sgrp 18656

This theorem is referenced by:  sgrppropd  18668  prdsplusgsgrpcl  18669  cntzsgrpcl  19275  rngpropd  20121