Theorem lidlss

Description: An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	lidlss.b	$$⊢B={\mathrm{Base}}_W$$
	lidlss.i	$$⊢I=\mathrm{LIdeal}\left(W\right)$$
Assertion	lidlss	$$⊢U\in I\to U\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		lidlss.b	$$⊢B={\mathrm{Base}}_W$$
2		lidlss.i	$$⊢I=\mathrm{LIdeal}\left(W\right)$$
3		rlmbas	$$⊢{\mathrm{Base}}_W={\mathrm{Base}}_{\mathrm{ringLMod}\left(W\right)}$$
4	1 3	eqtri	$$⊢B={\mathrm{Base}}_{\mathrm{ringLMod}\left(W\right)}$$
5		lidlval	$$⊢\mathrm{LIdeal}\left(W\right)=\mathrm{LSubSp}\left(\mathrm{ringLMod}\left(W\right)\right)$$
6	2 5	eqtri	$$⊢I=\mathrm{LSubSp}\left(\mathrm{ringLMod}\left(W\right)\right)$$
7	4 6	lssss	$$⊢U\in I\to U\subseteq B$$