Theorem cdleme21d

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 115, 3rd line. D , F , N , E , B , Z represent s_2, f(s), f_s(r), z_2, f(z), f_z(r) respectively. We prove f_s(r) = f_z(r). (Contributed by NM, 29-Nov-2012)

	Ref	Expression
Hypotheses	cdleme21.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme21.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme21.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme21.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme21.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme21.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme21.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme21.b	$$⊢B=\left(z\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme21.d	$$⊢D=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
	cdleme21.e	$$⊢E=\left(R\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W$$
	cdleme21d.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }D\right)$$
	cdleme21d.z	$$⊢Z=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(B\underset{\dot{}}{\vee }E\right)$$
Assertion	cdleme21d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to N=Z$$

Proof

Step	Hyp	Ref	Expression
1		cdleme21.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme21.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme21.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme21.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme21.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme21.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme21.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		cdleme21.b	$$⊢B=\left(z\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme21.d	$$⊢D=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
10		cdleme21.e	$$⊢E=\left(R\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W$$
11		cdleme21d.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }D\right)$$
12		cdleme21d.z	$$⊢Z=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(B\underset{\dot{}}{\vee }E\right)$$
13		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
14		simp12	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
15		simp13	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
16		simp2l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)$$
17		simp2r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)$$
18		simp33l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)$$
19		simp31	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to P\ne Q$$
20		simp11l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to K\in \mathrm{HL}$$
21		simp12l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to P\in A$$
22		simp13l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to Q\in A$$
23		simp2rl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to S\in A$$
24		simp32l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
25	18	simpld	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to z\in A$$
26		simp33r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z$$
27	1 2 4	cdleme21a	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\to S\ne z$$
28	20 21 22 23 24 25 26 27	syl322anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to S\ne z$$
29	19 28	jca	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to \left(P\ne Q\wedge S\ne z\right)$$
30	1 2 4	cdleme21b	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(S\in A\wedge P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\to \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
31	20 21 22 23 19 24 25 26 30	syl332anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
32		simp32r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
33	24 31 32	3jca	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
34	1 2 3 4 5 6	cdleme21c	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge Q\in A\right)\wedge \left(S\in A\wedge P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\to \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }z\right)$$
35	13 14 22 23 19 24 25 26 34	syl332anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }z\right)$$
36		eqid	$$⊢\left(S\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W=\left(S\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W$$
37	1 2 3 4 5 6 7 8 9 10 36 11 12	cdleme20	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\ne z\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }z\right)\right)\right)\to N=Z$$
38	13 14 15 16 17 18 29 33 35 37	syl333anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)\right)\to N=Z$$