Theorem grpsubcld

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

	Ref	Expression
Hypotheses	grpsubcld.b	$$⊢B={\mathrm{Base}}_G$$
	grpsubcld.m	$$⊢\underset{\dot{}}{-}={-}_G$$
	grpsubcld.g	$$⊢\phi \to G\in \mathrm{Grp}$$
	grpsubcld.x	$$⊢\phi \to X\in B$$
	grpsubcld.y	$$⊢\phi \to Y\in B$$
Assertion	grpsubcld	$$⊢\phi \to X\underset{\dot{}}{-}Y\in B$$

Proof

Step	Hyp	Ref	Expression
1		grpsubcld.b	$$⊢B={\mathrm{Base}}_G$$
2		grpsubcld.m	$$⊢\underset{\dot{}}{-}={-}_G$$
3		grpsubcld.g	$$⊢\phi \to G\in \mathrm{Grp}$$
4		grpsubcld.x	$$⊢\phi \to X\in B$$
5		grpsubcld.y	$$⊢\phi \to Y\in B$$
6	1 2	grpsubcl	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{-}Y\in B$$
7	3 4 5 6	syl3anc	$$⊢\phi \to X\underset{\dot{}}{-}Y\in B$$