Theorem mndcld

Description: Closure of the operation of a monoid. (Contributed by Thierry Arnoux, 3-Aug-2025)

	Ref	Expression
Hypotheses	mndcld.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
	mndcld.2	⊢ + = ( +g ‘ 𝐺 )
	mndcld.3	⊢ ( 𝜑 → 𝐺 ∈ Mnd )
	mndcld.4	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	mndcld.5	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	mndcld	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		mndcld.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		mndcld.2	⊢ + = ( +g ‘ 𝐺 )
3		mndcld.3	⊢ ( 𝜑 → 𝐺 ∈ Mnd )
4		mndcld.4	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		mndcld.5	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6	1 2	mndcl	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
7	3 4 5 6	syl3anc	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐵 )