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