Theorem rngocl

Description: Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ringi.1	$$⊢G=1^{st}\left(R\right)$$
	ringi.2	$$⊢H=2^{nd}\left(R\right)$$
	ringi.3	$$⊢X=\mathrm{ran}G$$
Assertion	rngocl	$$⊢\left(R\in \mathrm{RingOps}\wedge A\in X\wedge B\in X\right)\to AHB\in X$$

Proof

Step	Hyp	Ref	Expression
1		ringi.1	$$⊢G=1^{st}\left(R\right)$$
2		ringi.2	$$⊢H=2^{nd}\left(R\right)$$
3		ringi.3	$$⊢X=\mathrm{ran}G$$
4	1 2 3	rngosm	$$⊢R\in \mathrm{RingOps}\to H:X\times X⟶X$$
5		fovcdm	$$⊢\left(H:X\times X⟶X\wedge A\in X\wedge B\in X\right)\to AHB\in X$$
6	4 5	syl3an1	$$⊢\left(R\in \mathrm{RingOps}\wedge A\in X\wedge B\in X\right)\to AHB\in X$$