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 )