Theorem cdleme7a

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme7ga and cdleme7 . (Contributed by NM, 7-Jun-2012)

	Ref	Expression
Hypotheses	cdleme4.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme4.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme4.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme4.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme4.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme4.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme4.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme4.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme7.v	$$⊢V=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme7a	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }V\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme4.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme4.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme4.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme4.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme4.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme4.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme4.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
8		cdleme4.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme7.v	$$⊢V=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
10	9	oveq2i	$$⊢F\underset{\dot{}}{\vee }V=F\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)$$
11	10	oveq2i	$$⊢\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }V\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
12	8 11	eqtr4i	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }V\right)$$