Theorem grpcld

Description: Closure of the operation of a group. (Contributed by SN, 29-Jul-2024)

	Ref	Expression
Hypotheses	grpcld.b	$$⊢B={\mathrm{Base}}_G$$
	grpcld.p	$$⊢\underset{\dot{}}{+}={+}_G$$
	grpcld.r	$$⊢\phi \to G\in \mathrm{Grp}$$
	grpcld.x	$$⊢\phi \to X\in B$$
	grpcld.y	$$⊢\phi \to Y\in B$$
Assertion	grpcld	$$⊢\phi \to X\underset{\dot{}}{+}Y\in B$$

Proof

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