Theorem cdleme20c

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 114, second line. D , F , Y , G represent s_2, f(s), t_2, f(t). (Contributed by NM, 15-Nov-2012)

	Ref	Expression
Hypotheses	cdleme19.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme19.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme19.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme19.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme19.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme19.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme19.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme19.g	$$⊢G=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme19.d	$$⊢D=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
	cdleme19.y	$$⊢Y=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
	cdleme20.v	$$⊢V=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme20c	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to D\underset{\dot{}}{\vee }Y=\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$

Proof

Step	Hyp	Ref	Expression
1		cdleme19.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme19.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme19.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme19.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme19.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme19.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme19.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		cdleme19.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		cdleme19.d	$$⊢D=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
10		cdleme19.y	$$⊢Y=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
11		cdleme20.v	$$⊢V=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
12	9 10	oveq12i	$$⊢D\underset{\dot{}}{\vee }Y=\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)$$
13		simp1l	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to K\in \mathrm{HL}$$
14		simp21l	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\in A$$
15		simp22l	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to S\in A$$
16		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
17	16 2 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge R\in A\wedge S\in A\right)\to R\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
18	13 14 15 17	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
19		simp1r	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to W\in H$$
20	16 5	lhpbase	$$⊢W\in H\to W\in {\mathrm{Base}}_K$$
21	19 20	syl	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to W\in {\mathrm{Base}}_K$$
22	1 2 4	hlatlej1	$$⊢\left(K\in \mathrm{HL}\wedge R\in A\wedge S\in A\right)\to R\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }S\right)$$
23	13 14 15 22	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }S\right)$$
24	16 1 2 3 4	atmod2i1	$$⊢\left(K\in \mathrm{HL}\wedge \left(R\in A\wedge R\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\wedge R\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }S\right)\right)\to \left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }R=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(W\underset{\dot{}}{\vee }R\right)$$
25	13 14 18 21 23 24	syl131anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }R=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(W\underset{\dot{}}{\vee }R\right)$$
26		simp21	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)$$
27		eqid	$$⊢\mathrm{1.}\left(K\right)=\mathrm{1.}\left(K\right)$$
28	1 2 27 4 5	lhpjat1	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\to W\underset{\dot{}}{\vee }R=\mathrm{1.}\left(K\right)$$
29	13 19 26 28	syl21anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to W\underset{\dot{}}{\vee }R=\mathrm{1.}\left(K\right)$$
30	29	oveq2d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(W\underset{\dot{}}{\vee }R\right)=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\mathrm{1.}\left(K\right)$$
31		hlol	$$⊢K\in \mathrm{HL}\to K\in \mathrm{OL}$$
32	13 31	syl	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to K\in \mathrm{OL}$$
33	16 3 27	olm11	$$⊢\left(K\in \mathrm{OL}\wedge R\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\right)\to \left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\mathrm{1.}\left(K\right)=R\underset{\dot{}}{\vee }S$$
34	32 18 33	syl2anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\mathrm{1.}\left(K\right)=R\underset{\dot{}}{\vee }S$$
35	25 30 34	3eqtrrd	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\underset{\dot{}}{\vee }S=\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }R$$
36	35	oveq1d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }T=\left(\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T$$
37		simp22r	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg S\underset{\dot{}}{\le }W$$
38		simp3r	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
39		simp3l	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
40		eqid	$$⊢\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
41	1 2 3 4 5 40	cdlemeda	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\in A$$
42	13 19 15 37 14 38 39 41	syl223anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\in A$$
43		simp23	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to T\in A$$
44	2 4	hlatjass	$$⊢\left(K\in \mathrm{HL}\wedge \left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\in A\wedge R\in A\wedge T\in A\right)\right)\to \left(\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T=\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }T\right)$$
45	13 42 14 43 44	syl13anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T=\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }T\right)$$
46	36 45	eqtrd	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }T=\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }T\right)$$
47	46	oveq1d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W=\left(\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\wedge }W$$
48	16 2 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge R\in A\wedge T\in A\right)\to R\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
49	13 14 43 48	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
50	13	hllatd	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to K\in \mathrm{Lat}$$
51	16 1 3	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge R\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\to \left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
52	50 18 21 51	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
53	16 1 2 3 4	atmod1i1	$$⊢\left(K\in \mathrm{HL}\wedge \left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\in A\wedge R\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\wedge \left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W\right)\to \left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)=\left(\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\wedge }W$$
54	13 42 49 21 52 53	syl131anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)=\left(\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }T\right)\right)\underset{\dot{}}{\wedge }W$$
55	47 54	eqtr4d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W=\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)$$
56	12 55	eqtr4id	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to D\underset{\dot{}}{\vee }Y=\left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$