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 ) )