Theorem grpsubcld

Description: Closure of group subtraction. (Contributed by Thierry Arnoux, 3-Aug-2025)

	Ref	Expression
Hypotheses	grpsubcld.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpsubcld.m	⊢ − = ( -g ‘ 𝐺 )
	grpsubcld.g	⊢ ( 𝜑 → 𝐺 ∈ Grp )
	grpsubcld.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	grpsubcld.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	grpsubcld	⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		grpsubcld.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpsubcld.m	⊢ − = ( -g ‘ 𝐺 )
3		grpsubcld.g	⊢ ( 𝜑 → 𝐺 ∈ Grp )
4		grpsubcld.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		grpsubcld.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6	1 2	grpsubcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) ∈ 𝐵 )
7	3 4 5 6	syl3anc	⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ∈ 𝐵 )