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Metamath Proof Explorer

Theorem trlval5

Description: The value of the trace of a lattice translation in terms of itself. (Contributed by NM, 19-Jul-2013)

Ref Expression
Hypotheses trlval3.l 
|- .<_ = ( le ` K )
trlval3.j 
|- .\/ = ( join ` K )
trlval3.m 
|- ./\ = ( meet ` K )
trlval3.a 
|- A = ( Atoms ` K )
trlval3.h 
|- H = ( LHyp ` K )
trlval3.t 
|- T = ( ( LTrn ` K ) ` W )
trlval3.r 
|- R = ( ( trL ` K ) ` W )
Assertion trlval5 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P .\/ ( R ` F ) ) ./\ W ) )

Proof

Step Hyp Ref Expression
1 trlval3.l 
 |-  .<_ = ( le ` K )
2 trlval3.j 
 |-  .\/ = ( join ` K )
3 trlval3.m 
 |-  ./\ = ( meet ` K )
4 trlval3.a 
 |-  A = ( Atoms ` K )
5 trlval3.h 
 |-  H = ( LHyp ` K )
6 trlval3.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 trlval3.r 
 |-  R = ( ( trL ` K ) ` W )
8 1 2 3 4 5 6 7 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ W ) )
9 1 2 4 5 6 7 trljat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` F ) ) = ( P .\/ ( F ` P ) ) )
10 9 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( P .\/ ( R ` F ) ) ./\ W ) = ( ( P .\/ ( F ` P ) ) ./\ W ) )
11 8 10 eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P .\/ ( R ` F ) ) ./\ W ) )