Theorem cdleme6

Description: Part of proof of Lemma E in Crawley p. 113. This expresses (r \/ f_s(r)) /\ w = u at the top of p. 114. (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	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
Assertion	cdleme6	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge Q\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }W=U$$

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	1 2 3 4 5 6 7 8	cdleme5	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge Q\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\underset{\dot{}}{\vee }G=P\underset{\dot{}}{\vee }Q$$
10	9	oveq1d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge Q\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }W=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
11	10 6	eqtr4di	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge Q\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }W=U$$