Theorem cdleme3d

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme3fa and cdleme3 . (Contributed by NM, 6-Jun-2012)

	Ref	Expression
Hypotheses	cdleme1.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme1.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme1.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme1.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme1.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme1.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme1.f	$$⊢F=\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme3.3	$$⊢V=\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme3d	$$⊢F=\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }V\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme1.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme1.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme1.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme1.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme1.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme1.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme1.f	$$⊢F=\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
8		cdleme3.3	$$⊢V=\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W$$
9	8	oveq2i	$$⊢Q\underset{\dot{}}{\vee }V=Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)$$
10	9	oveq2i	$$⊢\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }V\right)=\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
11	7 10	eqtr4i	$$⊢F=\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }V\right)$$