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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 )