Theorem cdleme17a

Description: Part of proof of Lemma E in Crawley p. 114, first part of 4th paragraph. F , G , and C represent f(s), f_s(p), and s_1 respectively. We show, in their notation, f_s(p)=(p \/ q) /\ (q \/ s_1). (Contributed by NM, 11-Oct-2012)

	Ref	Expression
Hypotheses	cdleme17.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme17.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme17.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme17.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme17.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme17.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme17.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme17.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme17.c	$$⊢C=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme17a	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }C\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme17.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme17.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme17.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme17.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme17.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme17.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme17.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		cdleme17.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme17.c	$$⊢C=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
10	1 2 3 4 5 6 7 8 9	cdleme7a	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }C\right)$$
11	1 2 3 4 5 6 7 9	cdleme9	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to F\underset{\dot{}}{\vee }C=Q\underset{\dot{}}{\vee }C$$
12	11	oveq2d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }C\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }C\right)$$
13	10 12	eqtrid	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }C\right)$$