Theorem grpodivcl

Description: Closure of group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	grpdivf.1	⊢ 𝑋 = ran 𝐺
	grpdivf.3	⊢ 𝐷 = ( /𝑔 ‘ 𝐺 )
Assertion	grpodivcl	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		grpdivf.1	⊢ 𝑋 = ran 𝐺
2		grpdivf.3	⊢ 𝐷 = ( /𝑔 ‘ 𝐺 )
3	1 2	grpodivf	⊢ ( 𝐺 ∈ GrpOp → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 )
4		fovcdm	⊢ ( ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ 𝑋 )
5	3 4	syl3an1	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ 𝑋 )