Theorem mapdval5N

Description: Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	mapdval4.h	|- H = ( LHyp ` K )
	mapdval4.u	|- U = ( ( DVecH ` K ) ` W )
	mapdval4.s	|- S = ( LSubSp ` U )
	mapdval4.f	|- F = ( LFnl ` U )
	mapdval4.l	|- L = ( LKer ` U )
	mapdval4.o	|- O = ( ( ocH ` K ) ` W )
	mapdval4.m	|- M = ( ( mapd ` K ) ` W )
	mapdval4.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	mapdval4.t	|- ( ph -> T e. S )
Assertion	mapdval5N	|- ( ph -> ( M ` T ) = U_ v e. T { f e. F | ( O ` { v } ) = ( L ` f ) } )

Proof

Step	Hyp	Ref	Expression
1		mapdval4.h	|- H = ( LHyp ` K )
2		mapdval4.u	|- U = ( ( DVecH ` K ) ` W )
3		mapdval4.s	|- S = ( LSubSp ` U )
4		mapdval4.f	|- F = ( LFnl ` U )
5		mapdval4.l	|- L = ( LKer ` U )
6		mapdval4.o	|- O = ( ( ocH ` K ) ` W )
7		mapdval4.m	|- M = ( ( mapd ` K ) ` W )
8		mapdval4.k	|- ( ph -> ( K e. HL /\ W e. H ) )
9		mapdval4.t	|- ( ph -> T e. S )
10	1 2 3 4 5 6 7 8 9	mapdval4N	|- ( ph -> ( M ` T ) = { f e. F | E. v e. T ( O ` { v } ) = ( L ` f ) } )
11		iunrab	|- U_ v e. T { f e. F | ( O ` { v } ) = ( L ` f ) } = { f e. F | E. v e. T ( O ` { v } ) = ( L ` f ) }
12	10 11	eqtr4di	|- ( ph -> ( M ` T ) = U_ v e. T { f e. F | ( O ` { v } ) = ( L ` f ) } )