Metamath Proof Explorer


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 )