Theorem grpcl

Description: Closure of the operation of a group. (Contributed by NM, 14-Aug-2011)

	Ref	Expression
Hypotheses	grpcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpcl.p	⊢ + = ( +g ‘ 𝐺 )
Assertion	grpcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		grpcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpcl.p	⊢ + = ( +g ‘ 𝐺 )
3		grpmnd	⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd )
4	1 2	mndcl	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
5	3 4	syl3an1	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )