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Metamath Proof Explorer

Theorem grpcld

Description: Closure of the operation of a group. (Contributed by SN, 29-Jul-2024)

Ref Expression
Hypotheses grpcld.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
grpcld.p ⊢ + = ( +g ‘ 𝐺 )
grpcld.r ⊢ ( 𝜑 → 𝐺 ∈ Grp )
grpcld.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
grpcld.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion grpcld ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 grpcld.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 grpcld.p ⊢ + = ( +g ‘ 𝐺 )
3 grpcld.r ⊢ ( 𝜑 → 𝐺 ∈ Grp )
4 grpcld.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 grpcld.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6 1 2 grpcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
7 3 4 5 6 syl3anc ⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐵 )