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	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	trlval3.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	trlval3.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	trlval3.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	trlval3.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	trlval3.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
	trlval3.r	$$⊢R=\mathrm{trL}\left(K\right)\left(W\right)$$
Assertion	trlval5	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge F\in T\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\right)\to R\left(F\right)=\left(P\underset{\dot{}}{\vee }R\left(F\right)\right)\underset{\dot{}}{\wedge }W$$

Proof

Step	Hyp	Ref	Expression
1		trlval3.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		trlval3.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		trlval3.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		trlval3.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		trlval3.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		trlval3.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
7		trlval3.r	$$⊢R=\mathrm{trL}\left(K\right)\left(W\right)$$
8	1 2 3 4 5 6 7	trlval2	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge F\in T\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\right)\to R\left(F\right)=\left(P\underset{\dot{}}{\vee }F\left(P\right)\right)\underset{\dot{}}{\wedge }W$$
9	1 2 4 5 6 7	trljat1	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge F\in T\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\right)\to P\underset{\dot{}}{\vee }R\left(F\right)=P\underset{\dot{}}{\vee }F\left(P\right)$$
10	9	oveq1d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge F\in T\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\right)\to \left(P\underset{\dot{}}{\vee }R\left(F\right)\right)\underset{\dot{}}{\wedge }W=\left(P\underset{\dot{}}{\vee }F\left(P\right)\right)\underset{\dot{}}{\wedge }W$$
11	8 10	eqtr4d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge F\in T\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\right)\to R\left(F\right)=\left(P\underset{\dot{}}{\vee }R\left(F\right)\right)\underset{\dot{}}{\wedge }W$$