Theorem diael

Description: A member of the value of the partial isomorphism A is a translation, i.e., a vector. (Contributed by NM, 17-Jan-2014)

	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	diael	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ F e. ( I ` X ) ) -> F e. 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	1 2 3 4 5	diass	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) C_ T )
7	6	sseld	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( F e. ( I ` X ) -> F e. T ) )
8	7	3impia	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ F e. ( I ` X ) ) -> F e. T )