Theorem dalawlem14

Description: Lemma for dalaw . Combine dalawlem10 and dalawlem13 . (Contributed by NM, 6-Oct-2012)

	Ref	Expression
Hypotheses	dalawlem.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dalawlem.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	dalawlem.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalawlem.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalawlem2.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
Assertion	dalawlem14	$$⊢\left(\left(K\in \mathrm{HL}\wedge \neg \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		dalawlem.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		dalawlem.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		dalawlem.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		dalawlem.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		dalawlem2.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
6		ianor	$$⊢\neg \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\leftrightarrow \left(\neg \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\vee \neg \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)$$
7	1 2 3 4 5	dalawlem13	$$⊢\left(\left(K\in \mathrm{HL}\wedge \neg \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)$$
8	7	3expib	$$⊢\left(K\in \mathrm{HL}\wedge \neg \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
9	8	3exp	$$⊢K\in \mathrm{HL}\to \left(\neg \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\to \left(\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\right)\right)$$
10	1 2 3 4	dalawlem10	$$⊢\left(\left(K\in \mathrm{HL}\wedge \neg \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)$$
11	10	3expib	$$⊢\left(K\in \mathrm{HL}\wedge \neg \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
12	11	3exp	$$⊢K\in \mathrm{HL}\to \left(\neg \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\to \left(\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\right)\right)$$
13	9 12	jaod	$$⊢K\in \mathrm{HL}\to \left(\left(\neg \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\vee \neg \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\to \left(\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\right)\right)$$
14	6 13	biimtrid	$$⊢K\in \mathrm{HL}\to \left(\neg \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\to \left(\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)\right)\right)$$
15	14	3imp	$$⊢\left(K\in \mathrm{HL}\wedge \neg \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\to \left(\left(\left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
16	15	3impib	$$⊢\left(\left(K\in \mathrm{HL}\wedge \neg \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)\right)\right)$$