Metamath Proof Explorer


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 )