Theorem diass

Description: The value of the partial isomorphism A is a set of translations, i.e., a set of vectors. (Contributed by NM, 26-Nov-2013)

	Ref	Expression
Hypotheses	diass.b	|- B = ( Base ` K )
	diass.l	|- .<_ = ( le ` K )
	diass.h	|- H = ( LHyp ` K )
	diass.t	|- T = ( ( LTrn ` K ) ` W )
	diass.i	|- I = ( ( DIsoA ` K ) ` W )
Assertion	diass	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) C_ T )

Proof

Step	Hyp	Ref	Expression
1		diass.b	|- B = ( Base ` K )
2		diass.l	|- .<_ = ( le ` K )
3		diass.h	|- H = ( LHyp ` K )
4		diass.t	|- T = ( ( LTrn ` K ) ` W )
5		diass.i	|- I = ( ( DIsoA ` K ) ` W )
6		eqid	|- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
7	1 2 3 4 6 5	diaval	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = { f e. T | ( ( ( trL ` K ) ` W ) ` f ) .<_ X } )
8		ssrab2	|- { f e. T | ( ( ( trL ` K ) ` W ) ` f ) .<_ X } C_ T
9	7 8	eqsstrdi	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) C_ T )