Theorem cdleme15a

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, showing, in their notation, ((s \/ p) /\ (f(s) \/ q)) \/ ((t \/ p) /\ (f(t) \/ q))=((p \/ s_1) /\ (q \/ s_1)) \/ ((p \/ t_1) /\ (q \/ t_1)). We represent f(s), f(t), s_1, and t_1 with F , G , C , and X respectively. The order of our operations is slightly different. (Contributed by NM, 9-Oct-2012)

	Ref	Expression
Hypotheses	cdleme12.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme12.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme12.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme12.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme12.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme12.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme12.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme12.g	$$⊢G=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme15.c	$$⊢C=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
	cdleme15.x	$$⊢X=\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme15a	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(\left(T\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(G\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\right)\right)=\left(\left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }X\right)\right)\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }C\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }C\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme12.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme12.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme12.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme12.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme12.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme12.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme12.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
8		cdleme12.g	$$⊢G=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme15.c	$$⊢C=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
10		cdleme15.x	$$⊢X=\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
11		simp11l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to K\in \mathrm{HL}$$
12		simp11r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to W\in H$$
13		simp12l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to P\in A$$
14		simp12r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \neg P\underset{\dot{}}{\le }W$$
15		simp22l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to T\in A$$
16	1 2 3 4 5 10	cdleme8	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\to P\underset{\dot{}}{\vee }X=P\underset{\dot{}}{\vee }T$$
17	11 12 13 14 15 16	syl221anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to P\underset{\dot{}}{\vee }X=P\underset{\dot{}}{\vee }T$$
18	2 4	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge T\in A\right)\to P\underset{\dot{}}{\vee }T=T\underset{\dot{}}{\vee }P$$
19	11 13 15 18	syl3anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to P\underset{\dot{}}{\vee }T=T\underset{\dot{}}{\vee }P$$
20	17 19	eqtr2d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to T\underset{\dot{}}{\vee }P=P\underset{\dot{}}{\vee }X$$
21		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
22		simp12	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
23		simp13	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
24		simp22	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)$$
25		simp23l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to P\ne Q$$
26		simp32	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
27	1 2 3 4 5 6 8	cdleme3fa	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to G\in A$$
28	21 22 23 24 25 26 27	syl132anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to G\in A$$
29		simp13l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to Q\in A$$
30	2 4	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge G\in A\wedge Q\in A\right)\to G\underset{\dot{}}{\vee }Q=Q\underset{\dot{}}{\vee }G$$
31	11 28 29 30	syl3anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to G\underset{\dot{}}{\vee }Q=Q\underset{\dot{}}{\vee }G$$
32	1 2 3 4 5 6 10 6 8	cdleme11g	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge T\in A\right)\wedge P\ne Q\right)\to Q\underset{\dot{}}{\vee }G=Q\underset{\dot{}}{\vee }X$$
33	21 13 23 15 25 32	syl131anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to Q\underset{\dot{}}{\vee }G=Q\underset{\dot{}}{\vee }X$$
34	31 33	eqtrd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to G\underset{\dot{}}{\vee }Q=Q\underset{\dot{}}{\vee }X$$
35	20 34	oveq12d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(T\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(G\underset{\dot{}}{\vee }Q\right)=\left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }X\right)$$
36		simp21l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to S\in A$$
37	1 2 3 4 5 9	cdleme8	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge S\in A\right)\to P\underset{\dot{}}{\vee }C=P\underset{\dot{}}{\vee }S$$
38	11 12 13 14 36 37	syl221anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to P\underset{\dot{}}{\vee }C=P\underset{\dot{}}{\vee }S$$
39	38	eqcomd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to P\underset{\dot{}}{\vee }S=P\underset{\dot{}}{\vee }C$$
40	1 2 3 4 5 6 9 6 7	cdleme11g	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge S\in A\right)\wedge P\ne Q\right)\to Q\underset{\dot{}}{\vee }F=Q\underset{\dot{}}{\vee }C$$
41	21 13 23 36 25 40	syl131anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to Q\underset{\dot{}}{\vee }F=Q\underset{\dot{}}{\vee }C$$
42	39 41	oveq12d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\right)=\left(P\underset{\dot{}}{\vee }C\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }C\right)$$
43	35 42	oveq12d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(\left(T\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(G\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\right)\right)=\left(\left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }X\right)\right)\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }C\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }C\right)\right)$$