Theorem cdleme40m

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(x) is one-to-one on P .\/ Q line. TODO: FIX COMMENT Use proof idea from cdleme32sn1awN . (Contributed by NM, 18-Mar-2013)

	Ref	Expression
Hypotheses	cdleme40.b	$$⊢B={\mathrm{Base}}_K$$
	cdleme40.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme40.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme40.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme40.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme40.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme40.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme40.e	$$⊢E=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme40.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme40.i	$$⊢I=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=G\right)\right)$$
	cdleme40.n	$$⊢N=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,D\right)$$
	cdleme40a1.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme40a1.c	$$⊢C=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=Y\right)\right)$$
	cdleme40.t	$$⊢T=\left(v\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme40.f	$$⊢F=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }\left(\left(S\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
Assertion	cdleme40m	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to ⦋R/s⦌N\ne F$$

Proof

Step	Hyp	Ref	Expression
1		cdleme40.b	$$⊢B={\mathrm{Base}}_K$$
2		cdleme40.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdleme40.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdleme40.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdleme40.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdleme40.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdleme40.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdleme40.e	$$⊢E=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme40.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdleme40.i	$$⊢I=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=G\right)\right)$$
11		cdleme40.n	$$⊢N=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,D\right)$$
12		cdleme40a1.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
13		cdleme40a1.c	$$⊢C=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=Y\right)\right)$$
14		cdleme40.t	$$⊢T=\left(v\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
15		cdleme40.f	$$⊢F=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }\left(\left(S\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
16		simp22l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to R\in A$$
17		simp3l1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
18	9 10 11 12 13	cdleme31sn1c	$$⊢\left(R\in A\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to ⦋R/s⦌N=C$$
19	16 17 18	syl2anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to ⦋R/s⦌N=C$$
20	1	fvexi	$$⊢B\in \mathrm{V}$$
21		nfv	$$⊢Ⅎt\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)$$
22		nfra1	$$⊢Ⅎt\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=Y\right)$$
23		nfcv	$$⊢\underset{\_}{Ⅎ}tB$$
24	22 23	nfriota	$$⊢\underset{\_}{Ⅎ}t\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=Y\right)\right)$$
25	13 24	nfcxfr	$$⊢\underset{\_}{Ⅎ}tC$$
26		nfcv	$$⊢\underset{\_}{Ⅎ}tF$$
27	25 26	nfne	$$⊢ℲtC\ne F$$
28	27	a1i	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to ℲtC\ne F$$
29	13	a1i	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to C=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=Y\right)\right)$$
30		neeq1	$$⊢Y=C\to \left(Y\ne F\leftrightarrow C\ne F\right)$$
31	30	adantl	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\wedge Y=C\right)\to \left(Y\ne F\leftrightarrow C\ne F\right)$$
32		simpl1	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\wedge \left(t\in A\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)$$
33		simpl2	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\wedge \left(t\in A\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)$$
34		simpl3l	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\wedge \left(t\in A\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)$$
35		simprl	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\wedge \left(t\in A\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to t\in A$$
36		simprrl	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\wedge \left(t\in A\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg t\underset{\dot{}}{\le }W$$
37		simprrr	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\wedge \left(t\in A\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
38	35 36 37	jca31	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\wedge \left(t\in A\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
39		simp3r1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to v\in A$$
40		simp3r2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg v\underset{\dot{}}{\le }W$$
41		simp3r3	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
42	39 40 41	jca31	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(\left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\right)\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
43	42	adantr	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\wedge \left(t\in A\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(\left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\right)\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
44	2 3 4 5 6 7 8 12 14 15	cdleme39n	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\right)\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to Y\ne F$$
45	32 33 34 38 43 44	syl113anc	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\wedge \left(t\in A\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to Y\ne F$$
46	45	ex	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(\left(t\in A\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to Y\ne F\right)$$
47		simp1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)$$
48		simp22r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg R\underset{\dot{}}{\le }W$$
49		simp21	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to P\ne Q$$
50	1 2 3 4 5 6 7 8 12 13	cdleme25cl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to C\in B$$
51	47 16 48 49 17 50	syl122anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to C\in B$$
52		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
53		simp12	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
54		simp13	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
55	2 3 5 6	cdlemb2	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P\ne Q\right)\to \exists t\in A\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
56	52 53 54 49 55	syl121anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \exists t\in A\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
57	21 28 29 31 46 51 56	riotasv3d	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\wedge B\in \mathrm{V}\right)\to C\ne F$$
58	20 57	mpan2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to C\ne F$$
59	19 58	eqnetrd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to ⦋R/s⦌N\ne F$$