Theorem dibss

Description: The partial isomorphism B maps to a set of vectors in full vector space H. (Contributed by NM, 1-Jan-2014)

	Ref	Expression
Hypotheses	dibss.b	|- B = ( Base ` K )
	dibss.l	|- .<_ = ( le ` K )
	dibss.h	|- H = ( LHyp ` K )
	dibss.i	|- I = ( ( DIsoB ` K ) ` W )
	dibss.u	|- U = ( ( DVecH ` K ) ` W )
	dibss.v	|- V = ( Base ` U )
Assertion	dibss	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) C_ V )

Proof

Step	Hyp	Ref	Expression
1		dibss.b	|- B = ( Base ` K )
2		dibss.l	|- .<_ = ( le ` K )
3		dibss.h	|- H = ( LHyp ` K )
4		dibss.i	|- I = ( ( DIsoB ` K ) ` W )
5		dibss.u	|- U = ( ( DVecH ` K ) ` W )
6		dibss.v	|- V = ( Base ` U )
7		eqid	|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
8		eqid	|- ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
9	1 2 3 7 8	diass	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( DIsoA ` K ) ` W ) ` X ) C_ ( ( LTrn ` K ) ` W ) )
10		eqid	|- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
11		eqid	|- ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) )
12	1 3 7 10 11	tendo0cl	|- ( ( K e. HL /\ W e. H ) -> ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) e. ( ( TEndo ` K ) ` W ) )
13	12	snssd	|- ( ( K e. HL /\ W e. H ) -> { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } C_ ( ( TEndo ` K ) ` W ) )
14	13	adantr	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } C_ ( ( TEndo ` K ) ` W ) )
15		xpss12	|- ( ( ( ( ( DIsoA ` K ) ` W ) ` X ) C_ ( ( LTrn ` K ) ` W ) /\ { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } C_ ( ( TEndo ` K ) ` W ) ) -> ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) C_ ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
16	9 14 15	syl2anc	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) C_ ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
17	1 2 3 7 11 8 4	dibval2	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) )
18	3 7 10 5 6	dvhvbase	|- ( ( K e. HL /\ W e. H ) -> V = ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
19	18	adantr	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> V = ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) )
20	16 17 19	3sstr4d	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) C_ V )