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Metamath Proof Explorer

Theorem subgcld

Description: A subgroup is closed under group operation. (Contributed by Thierry Arnoux, 3-Jun-2025)

Ref Expression
Hypotheses subgcld.1 ⊢ + = ( +g ‘ 𝐺 )
subgcld.2 ⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
subgcld.3 ⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
subgcld.4 ⊢ ( 𝜑 → 𝑌 ∈ 𝑆 )
Assertion subgcld ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝑆 )

Proof

Step Hyp Ref Expression
1 subgcld.1 ⊢ + = ( +g ‘ 𝐺 )
2 subgcld.2 ⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
3 subgcld.3 ⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
4 subgcld.4 ⊢ ( 𝜑 → 𝑌 ∈ 𝑆 )
5 1 subgcl ⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )
6 2 3 4 5 syl3anc ⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝑆 )