Theorem cdleme20bN

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). We show v \/ s_2 = v \/ t_2. (Contributed by NM, 15-Nov-2012) (New usage is discouraged.)

	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	cdleme20bN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to V\underset{\dot{}}{\vee }D=V\underset{\dot{}}{\vee }Y$$

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		simp1l	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to K\in \mathrm{HL}$$
13	12	hllatd	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to K\in \mathrm{Lat}$$
14		simp22l	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to S\in A$$
15		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
16	15 4	atbase	$$⊢S\in A\to S\in {\mathrm{Base}}_K$$
17	14 16	syl	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to S\in {\mathrm{Base}}_K$$
18		simp21	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\in A$$
19	15 4	atbase	$$⊢R\in A\to R\in {\mathrm{Base}}_K$$
20	18 19	syl	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\in {\mathrm{Base}}_K$$
21		simp23l	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to T\in A$$
22	15 4	atbase	$$⊢T\in A\to T\in {\mathrm{Base}}_K$$
23	21 22	syl	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to T\in {\mathrm{Base}}_K$$
24	15 2	latj31	$$⊢\left(K\in \mathrm{Lat}\wedge \left(S\in {\mathrm{Base}}_K\wedge R\in {\mathrm{Base}}_K\wedge T\in {\mathrm{Base}}_K\right)\right)\to \left(S\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T=\left(T\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S$$
25	13 17 20 23 24	syl13anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(S\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T=\left(T\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S$$
26	25	oveq1d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(S\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W=\left(\left(T\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
27		simp1r	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to W\in H$$
28		simp22r	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg S\underset{\dot{}}{\le }W$$
29		simp31	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
30		simp33	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
31	1 2 3 4 5 6 7 8 9 10 11	cdleme20aN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to V\underset{\dot{}}{\vee }D=\left(\left(S\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
32	12 27 18 14 28 21 29 30 31	syl233anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to V\underset{\dot{}}{\vee }D=\left(\left(S\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
33	2 4	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge S\in A\wedge T\in A\right)\to S\underset{\dot{}}{\vee }T=T\underset{\dot{}}{\vee }S$$
34	12 14 21 33	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to S\underset{\dot{}}{\vee }T=T\underset{\dot{}}{\vee }S$$
35	34	oveq1d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W=\left(T\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
36	11 35	eqtrid	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to V=\left(T\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
37	36	oveq1d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to V\underset{\dot{}}{\vee }Y=\left(\left(T\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }Y$$
38		simp23r	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg T\underset{\dot{}}{\le }W$$
39		simp32	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
40		eqid	$$⊢\left(T\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W=\left(T\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
41	1 2 3 4 5 6 8 7 10 9 40	cdleme20aN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(T\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }Y=\left(\left(T\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
42	12 27 18 21 38 14 39 30 41	syl233anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(T\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }Y=\left(\left(T\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
43	37 42	eqtrd	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to V\underset{\dot{}}{\vee }Y=\left(\left(T\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
44	26 32 43	3eqtr4d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to V\underset{\dot{}}{\vee }D=V\underset{\dot{}}{\vee }Y$$