Theorem lcdsbase

Description: Base set of scalar ring for the closed kernel dual of a vector space. (Contributed by NM, 18-Mar-2015)

	Ref	Expression
Hypotheses	lcdsbase.h	|- H = ( LHyp ` K )
	lcdsbase.u	|- U = ( ( DVecH ` K ) ` W )
	lcdsbase.f	|- F = ( Scalar ` U )
	lcdsbase.l	|- L = ( Base ` F )
	lcdsbase.c	|- C = ( ( LCDual ` K ) ` W )
	lcdsbase.s	|- S = ( Scalar ` C )
	lcdsbase.r	|- R = ( Base ` S )
	lcdsbase.k	|- ( ph -> ( K e. HL /\ W e. H ) )
Assertion	lcdsbase	|- ( ph -> R = L )

Proof

Step	Hyp	Ref	Expression
1		lcdsbase.h	|- H = ( LHyp ` K )
2		lcdsbase.u	|- U = ( ( DVecH ` K ) ` W )
3		lcdsbase.f	|- F = ( Scalar ` U )
4		lcdsbase.l	|- L = ( Base ` F )
5		lcdsbase.c	|- C = ( ( LCDual ` K ) ` W )
6		lcdsbase.s	|- S = ( Scalar ` C )
7		lcdsbase.r	|- R = ( Base ` S )
8		lcdsbase.k	|- ( ph -> ( K e. HL /\ W e. H ) )
9		eqid	|- ( oppR ` F ) = ( oppR ` F )
10	1 2 3 9 5 6 8	lcdsca	|- ( ph -> S = ( oppR ` F ) )
11	10	fveq2d	|- ( ph -> ( Base ` S ) = ( Base ` ( oppR ` F ) ) )
12	9 4	opprbas	|- L = ( Base ` ( oppR ` F ) )
13	11 7 12	3eqtr4g	|- ( ph -> R = L )