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Metamath Proof Explorer

Theorem ltrnval1

Description: Value of a lattice translation under its co-atom. (Contributed by NM, 20-May-2012)

Ref Expression
Hypotheses ltrnval1.b 
|- B = ( Base ` K )
ltrnval1.l 
|- .<_ = ( le ` K )
ltrnval1.h 
|- H = ( LHyp ` K )
ltrnval1.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion ltrnval1 
|- ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ X .<_ W ) ) -> ( F ` X ) = X )

Proof

Step Hyp Ref Expression
1 ltrnval1.b 
 |-  B = ( Base ` K )
2 ltrnval1.l 
 |-  .<_ = ( le ` K )
3 ltrnval1.h 
 |-  H = ( LHyp ` K )
4 ltrnval1.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 eqid 
 |-  ( ( LDil ` K ) ` W ) = ( ( LDil ` K ) ` W )
6 3 5 4 ltrnldil 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T ) -> F e. ( ( LDil ` K ) ` W ) )
7 6 3adant3 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ X .<_ W ) ) -> F e. ( ( LDil ` K ) ` W ) )
8 1 2 3 5 ldilval 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. ( ( LDil ` K ) ` W ) /\ ( X e. B /\ X .<_ W ) ) -> ( F ` X ) = X )
9 7 8 syld3an2 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ X .<_ W ) ) -> ( F ` X ) = X )