Theorem lbsext

Description: For any linearly independent subset C of V , there is a basis containing the vectors in C . (Contributed by Mario Carneiro, 25-Jun-2014) (Revised by Mario Carneiro, 17-May-2015)

	Ref	Expression
Hypotheses	lbsex.j	|- J = ( LBasis ` W )
	lbsex.v	|- V = ( Base ` W )
	lbsex.n	|- N = ( LSpan ` W )
Assertion	lbsext	|- ( ( W e. LVec /\ C C_ V /\ A. x e. C -. x e. ( N ` ( C \ { x } ) ) ) -> E. s e. J C C_ s )

Proof

Step	Hyp	Ref	Expression
1		lbsex.j	|- J = ( LBasis ` W )
2		lbsex.v	|- V = ( Base ` W )
3		lbsex.n	|- N = ( LSpan ` W )
4	2	fvexi	|- V e. _V
5	4	pwex	|- ~P V e. _V
6		numth3	|- ( ~P V e. _V -> ~P V e. dom card )
7	5 6	ax-mp	|- ~P V e. dom card
8	7	jctr	|- ( W e. LVec -> ( W e. LVec /\ ~P V e. dom card ) )
9	1 2 3	lbsextg	|- ( ( ( W e. LVec /\ ~P V e. dom card ) /\ C C_ V /\ A. x e. C -. x e. ( N ` ( C \ { x } ) ) ) -> E. s e. J C C_ s )
10	8 9	syl3an1	|- ( ( W e. LVec /\ C C_ V /\ A. x e. C -. x e. ( N ` ( C \ { x } ) ) ) -> E. s e. J C C_ s )