Theorem dimcl

Description: Closure of the vector space dimension. (Contributed by Thierry Arnoux, 18-May-2023)

	Ref	Expression
Assertion	dimcl	|- ( V e. LVec -> ( dim ` V ) e. NN0* )

Proof

Step	Hyp	Ref	Expression
1		eqid	|- ( LBasis ` V ) = ( LBasis ` V )
2	1	lbsex	|- ( V e. LVec -> ( LBasis ` V ) =/= (/) )
3		n0	|- ( ( LBasis ` V ) =/= (/) <-> E. b b e. ( LBasis ` V ) )
4	2 3	sylib	|- ( V e. LVec -> E. b b e. ( LBasis ` V ) )
5	1	dimval	|- ( ( V e. LVec /\ b e. ( LBasis ` V ) ) -> ( dim ` V ) = ( # ` b ) )
6		hashxnn0	|- ( b e. ( LBasis ` V ) -> ( # ` b ) e. NN0* )
7	6	adantl	|- ( ( V e. LVec /\ b e. ( LBasis ` V ) ) -> ( # ` b ) e. NN0* )
8	5 7	eqeltrd	|- ( ( V e. LVec /\ b e. ( LBasis ` V ) ) -> ( dim ` V ) e. NN0* )
9	4 8	exlimddv	|- ( V e. LVec -> ( dim ` V ) e. NN0* )