Theorem lbslinds

Description: A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015)

	Ref	Expression
Hypothesis	lbslinds.j	$$⊢J=\mathrm{LBasis}\left(W\right)$$
Assertion	lbslinds	$$⊢J\subseteq \mathrm{LIndS}\left(W\right)$$

Proof

Step	Hyp	Ref	Expression
1		lbslinds.j	$$⊢J=\mathrm{LBasis}\left(W\right)$$
2		eqid	$$⊢{\mathrm{Base}}_W={\mathrm{Base}}_W$$
3		eqid	$$⊢\mathrm{LSpan}\left(W\right)=\mathrm{LSpan}\left(W\right)$$
4	2 1 3	islbs4	$$⊢a\in J\leftrightarrow \left(a\in \mathrm{LIndS}\left(W\right)\wedge \mathrm{LSpan}\left(W\right)\left(a\right)={\mathrm{Base}}_W\right)$$
5	4	simplbi	$$⊢a\in J\to a\in \mathrm{LIndS}\left(W\right)$$
6	5	ssriv	$$⊢J\subseteq \mathrm{LIndS}\left(W\right)$$