Theorem mapdval3N

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

	Ref	Expression
Hypotheses	mapdval2.h	|- H = ( LHyp ` K )
	mapdval2.u	|- U = ( ( DVecH ` K ) ` W )
	mapdval2.s	|- S = ( LSubSp ` U )
	mapdval2.n	|- N = ( LSpan ` U )
	mapdval2.f	|- F = ( LFnl ` U )
	mapdval2.l	|- L = ( LKer ` U )
	mapdval2.o	|- O = ( ( ocH ` K ) ` W )
	mapdval2.m	|- M = ( ( mapd ` K ) ` W )
	mapdval2.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	mapdval2.t	|- ( ph -> T e. S )
	mapdval2.c	|- C = { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) }
Assertion	mapdval3N	|- ( ph -> ( M ` T ) = U_ v e. T { f e. C | ( O ` ( L ` f ) ) = ( N ` { v } ) } )

Proof

Step	Hyp	Ref	Expression
1		mapdval2.h	|- H = ( LHyp ` K )
2		mapdval2.u	|- U = ( ( DVecH ` K ) ` W )
3		mapdval2.s	|- S = ( LSubSp ` U )
4		mapdval2.n	|- N = ( LSpan ` U )
5		mapdval2.f	|- F = ( LFnl ` U )
6		mapdval2.l	|- L = ( LKer ` U )
7		mapdval2.o	|- O = ( ( ocH ` K ) ` W )
8		mapdval2.m	|- M = ( ( mapd ` K ) ` W )
9		mapdval2.k	|- ( ph -> ( K e. HL /\ W e. H ) )
10		mapdval2.t	|- ( ph -> T e. S )
11		mapdval2.c	|- C = { g e. F | ( O ` ( O ` ( L ` g ) ) ) = ( L ` g ) }
12	1 2 3 4 5 6 7 8 9 10 11	mapdval2N	|- ( ph -> ( M ` T ) = { f e. C | E. v e. T ( O ` ( L ` f ) ) = ( N ` { v } ) } )
13		iunrab	|- U_ v e. T { f e. C | ( O ` ( L ` f ) ) = ( N ` { v } ) } = { f e. C | E. v e. T ( O ` ( L ` f ) ) = ( N ` { v } ) }
14	12 13	eqtr4di	|- ( ph -> ( M ` T ) = U_ v e. T { f e. C | ( O ` ( L ` f ) ) = ( N ` { v } ) } )