Theorem dalawlem1

Description: Lemma for dalaw . Special case of dath2 , where C is replaced by ( ( P .\/ S ) ./\ ( Q .\/ T ) ) . The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 . (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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalawlem.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
Assertion	dalawlem1	$$⊢\left(\left(K\in \mathrm{HL}\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)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\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(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\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(T\underset{\dot{}}{\vee }U\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(U\underset{\dot{}}{\vee }S\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)\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		dalawlem.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
6		simp11	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to K\in \mathrm{HL}$$
7	6	hllatd	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to K\in \mathrm{Lat}$$
8		simp121	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to P\in A$$
9		simp131	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to S\in A$$
10		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
11	10 2 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge S\in A\right)\to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
12	6 8 9 11	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
13		simp122	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to Q\in A$$
14		simp132	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to T\in A$$
15	10 2 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge Q\in A\wedge T\in A\right)\to Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
16	6 13 14 15	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
17	10 3	latmcl	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\wedge Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\right)\to \left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\in {\mathrm{Base}}_K$$
18	7 12 16 17	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\in {\mathrm{Base}}_K$$
19	6 18	jca	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge \left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\in {\mathrm{Base}}_K\right)$$
20		simp12	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to \left(P\in A\wedge Q\in A\wedge R\in A\right)$$
21		simp13	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to \left(S\in A\wedge T\in A\wedge U\in A\right)$$
22		simp2l	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O$$
23		simp2r	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O$$
24		simp31	$$⊢\left(\left(K\in \mathrm{HL}\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)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\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(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\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(T\underset{\dot{}}{\vee }U\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(U\underset{\dot{}}{\vee }S\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)\right)\to \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)$$
25		simp32	$$⊢\left(\left(K\in \mathrm{HL}\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)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\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(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\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(T\underset{\dot{}}{\vee }U\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(U\underset{\dot{}}{\vee }S\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)\right)\to \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(S\underset{\dot{}}{\vee }T\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(T\underset{\dot{}}{\vee }U\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(U\underset{\dot{}}{\vee }S\right)\right)$$
26	10 1 3	latmle1	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\wedge Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\right)\to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
27	7 12 16 26	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
28	10 1 3	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\wedge Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\right)\to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
29	7 12 16 28	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
30		simp33	$$⊢\left(\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)$$
31	27 29 30	3jca	$$⊢\left(\left(K\in \mathrm{HL}\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)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\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(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\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(T\underset{\dot{}}{\vee }U\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(U\underset{\dot{}}{\vee }S\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)\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(P\underset{\dot{}}{\vee }S\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(Q\underset{\dot{}}{\vee }T\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)$$
32		eqid	$$⊢\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
33		eqid	$$⊢\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)$$
34		eqid	$$⊢\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)=\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(U\underset{\dot{}}{\vee }S\right)$$
35	10 1 2 4 3 5 32 33 34	dath2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge \left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\in {\mathrm{Base}}_K\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)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\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(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\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(T\underset{\dot{}}{\vee }U\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(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\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 }S\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(Q\underset{\dot{}}{\vee }T\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)\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)$$
36	19 20 21 22 23 24 25 31 35	syl323anc	$$⊢\left(\left(K\in \mathrm{HL}\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)\wedge \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\in O\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\in O\right)\wedge \left(\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(\neg \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\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(T\underset{\dot{}}{\vee }U\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(U\underset{\dot{}}{\vee }S\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)\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)$$