Theorem sgrpcl

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

	Ref	Expression
Hypotheses	sgrpass.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	sgrpass.o	⊢ ⚬ = ( +g ‘ 𝐺 )
Assertion	sgrpcl	⊢ ( ( 𝐺 ∈ Smgrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ⚬ 𝑌 ) ∈ 𝐵 )

Proof

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