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Metamath Proof Explorer

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 )