Theorem dih1cnv

Description: The isomorphism H converse value of the full vector space is the lattice one. (Contributed by NM, 19-Jun-2014)

	Ref	Expression
Hypotheses	dih1cnv.h	|- H = ( LHyp ` K )
	dih1cnv.m	|- .1. = ( 1. ` K )
	dih1cnv.i	|- I = ( ( DIsoH ` K ) ` W )
	dih1cnv.u	|- U = ( ( DVecH ` K ) ` W )
	dih1cnv.v	|- V = ( Base ` U )
Assertion	dih1cnv	|- ( ( K e. HL /\ W e. H ) -> ( `' I ` V ) = .1. )

Proof

Step	Hyp	Ref	Expression
1		dih1cnv.h	|- H = ( LHyp ` K )
2		dih1cnv.m	|- .1. = ( 1. ` K )
3		dih1cnv.i	|- I = ( ( DIsoH ` K ) ` W )
4		dih1cnv.u	|- U = ( ( DVecH ` K ) ` W )
5		dih1cnv.v	|- V = ( Base ` U )
6	2 1 3 4 5	dih1	|- ( ( K e. HL /\ W e. H ) -> ( I ` .1. ) = V )
7	6	fveq2d	|- ( ( K e. HL /\ W e. H ) -> ( `' I ` ( I ` .1. ) ) = ( `' I ` V ) )
8		hlop	|- ( K e. HL -> K e. OP )
9	8	adantr	|- ( ( K e. HL /\ W e. H ) -> K e. OP )
10		eqid	|- ( Base ` K ) = ( Base ` K )
11	10 2	op1cl	|- ( K e. OP -> .1. e. ( Base ` K ) )
12	9 11	syl	|- ( ( K e. HL /\ W e. H ) -> .1. e. ( Base ` K ) )
13	10 1 3	dihcnvid1	|- ( ( ( K e. HL /\ W e. H ) /\ .1. e. ( Base ` K ) ) -> ( `' I ` ( I ` .1. ) ) = .1. )
14	12 13	mpdan	|- ( ( K e. HL /\ W e. H ) -> ( `' I ` ( I ` .1. ) ) = .1. )
15	7 14	eqtr3d	|- ( ( K e. HL /\ W e. H ) -> ( `' I ` V ) = .1. )