Theorem sgrpcl 18655

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 18653	. 2 ⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
2		sgrpass.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3		sgrpass.o	. . 3 ⊢ ⚬ = (+g‘𝐺)
4	2, 3	mgmcl 18572	. 2 ⊢ ((𝐺 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)
5	1, 4	syl3an1 1160	1 ⊢ ((𝐺 ∈ Smgrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ‘cfv 6534  (class class class)co 7402  Basecbs 17149  +_gcplusg 17202  Mgmcmgm 18567  Smgrpcsgrp 18647

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 2695  ax-nul 5297

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 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-ov 7405  df-mgm 18569  df-sgrp 18648

This theorem is referenced by:  sgrppropd  18660  prdsplusgsgrpcl  18661  cntzsgrpcl  19246  rngpropd  20075