Theorem subgsubcld

Description: A subgroup is closed under group subtraction. (Contributed by Thierry Arnoux, 6-Jul-2025)

	Ref	Expression
Hypotheses	subgsubcld.m	⊢ − = ( -g ‘ 𝐺 )
	subgsubcld.s	⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
	subgsubcld.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
	subgsubcld.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑆 )
Assertion	subgsubcld	⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		subgsubcld.m	⊢ − = ( -g ‘ 𝐺 )
2		subgsubcld.s	⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
3		subgsubcld.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
4		subgsubcld.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑆 )
5	1	subgsubcl	⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑋 − 𝑌 ) ∈ 𝑆 )
6	2 3 4 5	syl3anc	⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ∈ 𝑆 )