Theorem cdleme16g

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, Eq. (1). F and G represent f(s) and f(t) respectively. We show, in their notation, (s \/ t) /\ w=(f(s) \/ f(t)) /\ w. (Contributed by NM, 11-Oct-2012)

	Ref	Expression
Hypotheses	cdleme12.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme12.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme12.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme12.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme12.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme12.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme12.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme12.g	$$⊢G=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
Assertion	cdleme16g	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W=\left(F\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }W$$

Proof

Step	Hyp	Ref	Expression
1		cdleme12.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme12.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme12.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme12.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme12.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme12.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme12.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		cdleme12.g	$$⊢G=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9	1 2 3 4 5 6 7 8	cdleme16e	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }G\right)=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
10	1 2 3 4 5 6 7 8	cdleme16f	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }G\right)=\left(F\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }W$$
11	9 10	eqtr3d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W=\left(F\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }W$$