Theorem mndcl

Description: Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015) (Proof shortened by AV, 8-Feb-2020)

	Ref	Expression
Hypotheses	mndcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	mndcl.p	⊢ + = ( +g ‘ 𝐺 )
Assertion	mndcl	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		mndcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		mndcl.p	⊢ + = ( +g ‘ 𝐺 )
3		mndmgm	⊢ ( 𝐺 ∈ Mnd → 𝐺 ∈ Mgm )
4	1 2	mgmcl	⊢ ( ( 𝐺 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
5	3 4	syl3an1	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )