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Metamath Proof Explorer

Theorem ltrnj

Description: Lattice translation of a meet. TODO: change antecedent to K e. HL (Contributed by NM, 25-May-2012)

Ref Expression
Hypotheses ltrnj.b 
|- B = ( Base ` K )
ltrnj.j 
|- .\/ = ( join ` K )
ltrnj.h 
|- H = ( LHyp ` K )
ltrnj.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion ltrnj 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> ( F ` ( X .\/ Y ) ) = ( ( F ` X ) .\/ ( F ` Y ) ) )

Proof

Step Hyp Ref Expression
1 ltrnj.b 
 |-  B = ( Base ` K )
2 ltrnj.j 
 |-  .\/ = ( join ` K )
3 ltrnj.h 
 |-  H = ( LHyp ` K )
4 ltrnj.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> K e. HL )
6 5 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> K e. Lat )
7 eqid 
 |-  ( LAut ` K ) = ( LAut ` K )
8 3 7 4 ltrnlaut 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> F e. ( LAut ` K ) )
9 8 3adant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> F e. ( LAut ` K ) )
10 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> X e. B )
11 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> Y e. B )
12 1 2 7 lautj 
 |-  ( ( K e. Lat /\ ( F e. ( LAut ` K ) /\ X e. B /\ Y e. B ) ) -> ( F ` ( X .\/ Y ) ) = ( ( F ` X ) .\/ ( F ` Y ) ) )
13 6 9 10 11 12 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> ( F ` ( X .\/ Y ) ) = ( ( F ` X ) .\/ ( F ` Y ) ) )