Theorem dih1rn

Description: The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014)

	Ref	Expression
Hypotheses	dih1rn.h	|- H = ( LHyp ` K )
	dih1rn.i	|- I = ( ( DIsoH ` K ) ` W )
	dih1rn.u	|- U = ( ( DVecH ` K ) ` W )
	dih1rn.v	|- V = ( Base ` U )
Assertion	dih1rn	|- ( ( K e. HL /\ W e. H ) -> V e. ran I )

Proof

Step	Hyp	Ref	Expression
1		dih1rn.h	|- H = ( LHyp ` K )
2		dih1rn.i	|- I = ( ( DIsoH ` K ) ` W )
3		dih1rn.u	|- U = ( ( DVecH ` K ) ` W )
4		dih1rn.v	|- V = ( Base ` U )
5		eqid	|- ( 1. ` K ) = ( 1. ` K )
6	5 1 2 3 4	dih1	|- ( ( K e. HL /\ W e. H ) -> ( I ` ( 1. ` K ) ) = V )
7		hlop	|- ( K e. HL -> K e. OP )
8	7	adantr	|- ( ( K e. HL /\ W e. H ) -> K e. OP )
9		eqid	|- ( Base ` K ) = ( Base ` K )
10	9 5	op1cl	|- ( K e. OP -> ( 1. ` K ) e. ( Base ` K ) )
11	8 10	syl	|- ( ( K e. HL /\ W e. H ) -> ( 1. ` K ) e. ( Base ` K ) )
12	9 1 2	dihcl	|- ( ( ( K e. HL /\ W e. H ) /\ ( 1. ` K ) e. ( Base ` K ) ) -> ( I ` ( 1. ` K ) ) e. ran I )
13	11 12	mpdan	|- ( ( K e. HL /\ W e. H ) -> ( I ` ( 1. ` K ) ) e. ran I )
14	6 13	eqeltrrd	|- ( ( K e. HL /\ W e. H ) -> V e. ran I )