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	$$⊢X=\mathrm{ran}G$$
	grpdivf.3	$$⊢D={/}_g\left(G\right)$$
Assertion	grpodivcl	$$⊢\left(G\in \mathrm{GrpOp}\wedge A\in X\wedge B\in X\right)\to ADB\in X$$

Proof

Step	Hyp	Ref	Expression
1		grpdivf.1	$$⊢X=\mathrm{ran}G$$
2		grpdivf.3	$$⊢D={/}_g\left(G\right)$$
3	1 2	grpodivf	$$⊢G\in \mathrm{GrpOp}\to D:X\times X⟶X$$
4		fovcdm	$$⊢\left(D:X\times X⟶X\wedge A\in X\wedge B\in X\right)\to ADB\in X$$
5	3 4	syl3an1	$$⊢\left(G\in \mathrm{GrpOp}\wedge A\in X\wedge B\in X\right)\to ADB\in X$$