Theorem grpsubcl

Description: Closure of group subtraction. (Contributed by NM, 31-Mar-2014)

	Ref	Expression
Hypotheses	grpsubcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpsubcl.m	⊢ − = ( -g ‘ 𝐺 )
Assertion	grpsubcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		grpsubcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpsubcl.m	⊢ − = ( -g ‘ 𝐺 )
3	1 2	grpsubf	⊢ ( 𝐺 ∈ Grp → − : ( 𝐵 × 𝐵 ) ⟶ 𝐵 )
4		fovrn	⊢ ( ( − : ( 𝐵 × 𝐵 ) ⟶ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) ∈ 𝐵 )
5	3 4	syl3an1	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) ∈ 𝐵 )