Theorem cdleme35g

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013)

	Ref	Expression
Hypotheses	cdleme35.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme35.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme35.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme35.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme35.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme35.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme35.f	$$⊢F=\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
Assertion	cdleme35g	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(F\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }F\right)\underset{\dot{}}{\wedge }W\right)\right)=R$$

Proof

Step	Hyp	Ref	Expression
1		cdleme35.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme35.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme35.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme35.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme35.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme35.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme35.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	1 2 3 4 5 6 7	cdleme35a	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to F\underset{\dot{}}{\vee }U=R\underset{\dot{}}{\vee }U$$
9	1 2 3 4 5 6 7	cdleme35e	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }F\right)\underset{\dot{}}{\wedge }W\right)=P\underset{\dot{}}{\vee }R$$
10	8 9	oveq12d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(F\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }F\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }R\right)$$
11	1 2 3 4 5 6 7	cdleme35f	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }R\right)=R$$
12	10 11	eqtrd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(F\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }F\right)\underset{\dot{}}{\wedge }W\right)\right)=R$$