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Metamath Proof Explorer

Theorem submcld

Description: Submonoids are closed under the monoid operation. (Contributed by Thierry Arnoux, 4-May-2025)

Ref Expression
Hypotheses submcld.1 ⊢ + = ( +g ‘ 𝑀 )
submcld.2 ⊢ ( 𝜑 → 𝑆 ∈ ( SubMnd ‘ 𝑀 ) )
submcld.3 ⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
submcld.4 ⊢ ( 𝜑 → 𝑌 ∈ 𝑆 )
Assertion submcld ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝑆 )

Proof

Step Hyp Ref Expression
1 submcld.1 ⊢ + = ( +g ‘ 𝑀 )
2 submcld.2 ⊢ ( 𝜑 → 𝑆 ∈ ( SubMnd ‘ 𝑀 ) )
3 submcld.3 ⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
4 submcld.4 ⊢ ( 𝜑 → 𝑌 ∈ 𝑆 )
5 1 submcl ⊢ ( ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )
6 2 3 4 5 syl3anc ⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝑆 )