Theorem cdleme26e

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph, 4th line on p. 115. F , N , O represent f(z), f_z(s), f_z(t) respectively. When t \/ v = p \/ q, f_z(s) <_ f_z(t) \/ v. TODO: FIX COMMENT. (Contributed by NM, 2-Feb-2013)

	Ref	Expression
Hypotheses	cdleme26.b	$$⊢B={\mathrm{Base}}_K$$
	cdleme26.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme26.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme26.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme26.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme26.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme26e.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme26e.f	$$⊢F=\left(z\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme26e.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(S\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme26e.o	$$⊢O=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(T\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme26e.i	$$⊢I=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)$$
	cdleme26e.e	$$⊢E=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=O\right)\right)$$
Assertion	cdleme26e	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to I\underset{\dot{}}{\le }\left(E\underset{\dot{}}{\vee }V\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme26.b	$$⊢B={\mathrm{Base}}_K$$
2		cdleme26.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdleme26.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdleme26.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdleme26.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdleme26.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdleme26e.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdleme26e.f	$$⊢F=\left(z\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme26e.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(S\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdleme26e.o	$$⊢O=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(T\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
11		cdleme26e.i	$$⊢I=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)$$
12		cdleme26e.e	$$⊢E=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=O\right)\right)$$
13		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
14		simp12	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
15		simp13	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
16		simp21l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to S\in A$$
17		simp22l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to T\in A$$
18	16 17	jca	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to \left(S\in A\wedge T\in A\right)$$
19		simp23	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)$$
20		simp311	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to P\ne Q$$
21		simp32l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q$$
22	20 21	jca	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to \left(P\ne Q\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)$$
23		simp33	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)$$
24	2 3 4 5 6 7 8 9 10	cdleme22e	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge T\in A\right)\right)\wedge \left(\left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to N\underset{\dot{}}{\le }\left(O\underset{\dot{}}{\vee }V\right)$$
25	13 14 15 18 19 22 23 24	syl133anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to N\underset{\dot{}}{\le }\left(O\underset{\dot{}}{\vee }V\right)$$
26		simp21r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to \neg S\underset{\dot{}}{\le }W$$
27		simp312	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
28	1 2 3 4 5 6 7 8 9 11	cdleme25cl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to I\in B$$
29	13 14 15 16 26 20 27 28	syl322anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to I\in B$$
30		simp33l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to z\in A$$
31		simp33r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to \neg z\underset{\dot{}}{\le }W$$
32		simp32r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
33	31 32	jca	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to \left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
34	1	fvexi	$$⊢B\in \mathrm{V}$$
35	34 11	riotasv	$$⊢\left(I\in B\wedge z\in A\wedge \left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to I=N$$
36	29 30 33 35	syl3anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to I=N$$
37		simp22r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to \neg T\underset{\dot{}}{\le }W$$
38		simp313	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
39	1 2 3 4 5 6 7 8 10 12	cdleme25cl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to E\in B$$
40	13 14 15 17 37 20 38 39	syl322anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to E\in B$$
41	34 12	riotasv	$$⊢\left(E\in B\wedge z\in A\wedge \left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to E=O$$
42	40 30 33 41	syl3anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to E=O$$
43	42	oveq1d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to E\underset{\dot{}}{\vee }V=O\underset{\dot{}}{\vee }V$$
44	25 36 43	3brtr4d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(V\in A\wedge V\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(T\underset{\dot{}}{\vee }V=P\underset{\dot{}}{\vee }Q\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\right)\right)\to I\underset{\dot{}}{\le }\left(E\underset{\dot{}}{\vee }V\right)$$