Theorem cdleme40n

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. TODO get rid of '.<' class? (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	$$⊢F=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }\left(\left(S\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme40a1.x	$$⊢X=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme40.o	$$⊢O=\left(\iota z\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=X\right)\right)$$
	cdleme40.v	$$⊢V=if\left(u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),O,\underset{\dot{}}{<}\right)$$
	cdleme40a1.z	$$⊢Z=\left(\iota z\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=F\right)\right)$$
Assertion	cdleme40n	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to ⦋R/s⦌N\ne ⦋S/u⦌V$$

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		cdleme40a1.x	$$⊢X=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
17		cdleme40.o	$$⊢O=\left(\iota z\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=X\right)\right)$$
18		cdleme40.v	$$⊢V=if\left(u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),O,\underset{\dot{}}{<}\right)$$
19		cdleme40a1.z	$$⊢Z=\left(\iota z\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=F\right)\right)$$
20	1	fvexi	$$⊢B\in \mathrm{V}$$
21		nfv	$$⊢Ⅎv\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)$$
22		nfcv	$$⊢\underset{\_}{Ⅎ}v⦋R/s⦌N$$
23		nfra1	$$⊢Ⅎv\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=F\right)$$
24		nfcv	$$⊢\underset{\_}{Ⅎ}vB$$
25	23 24	nfriota	$$⊢\underset{\_}{Ⅎ}v\left(\iota z\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=F\right)\right)$$
26	19 25	nfcxfr	$$⊢\underset{\_}{Ⅎ}vZ$$
27	22 26	nfne	$$⊢Ⅎv⦋R/s⦌N\ne Z$$
28	27	a1i	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to Ⅎv⦋R/s⦌N\ne Z$$
29	19	a1i	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to Z=\left(\iota z\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=F\right)\right)$$
30		neeq2	$$⊢F=Z\to \left(⦋R/s⦌N\ne F\leftrightarrow ⦋R/s⦌N\ne Z\right)$$
31	30	adantl	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge F=Z\right)\to \left(⦋R/s⦌N\ne F\leftrightarrow ⦋R/s⦌N\ne Z\right)$$
32		simpl11	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge \left(v\in A\wedge \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
33		simpl12	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge \left(v\in A\wedge \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
34		simpl13	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge \left(v\in A\wedge \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
35		simpl21	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge \left(v\in A\wedge \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to P\ne Q$$
36		simpl22	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge \left(v\in A\wedge \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)$$
37		simpl23	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge \left(v\in A\wedge \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)$$
38		simpl3	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge \left(v\in A\wedge \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)$$
39		simprl	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge \left(v\in A\wedge \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to v\in A$$
40		simprrl	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge \left(v\in A\wedge \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg v\underset{\dot{}}{\le }W$$
41		simprrr	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge \left(v\in A\wedge \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
42	39 40 41	3jca	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge \left(v\in A\wedge \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
43	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15	cdleme40m	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\wedge \left(v\in A\wedge \neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to ⦋R/s⦌N\ne F$$
44	32 33 34 35 36 37 38 42 43	syl332anc	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge \left(v\in A\wedge \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to ⦋R/s⦌N\ne F$$
45	44	ex	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to \left(\left(v\in A\wedge \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to ⦋R/s⦌N\ne F\right)$$
46		simp1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to \left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)$$
47		simp23l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to S\in A$$
48		simp23r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to \neg S\underset{\dot{}}{\le }W$$
49		simp21	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to P\ne Q$$
50		simp32	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
51	1 2 3 4 5 6 7 14 15 19	cdleme25cl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to Z\in B$$
52	46 47 48 49 50 51	syl122anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to Z\in B$$
53		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
54		simp12	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
55		simp13	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
56	2 3 5 6	cdlemb2	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P\ne Q\right)\to \exists v\in A\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
57	53 54 55 49 56	syl121anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to \exists v\in A\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
58	21 28 29 31 45 52 57	riotasv3d	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\wedge B\in \mathrm{V}\right)\to ⦋R/s⦌N\ne Z$$
59	20 58	mpan2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to ⦋R/s⦌N\ne Z$$
60	16 17 18 15 19	cdleme31sn1c	$$⊢\left(S\in A\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to ⦋S/u⦌V=Z$$
61	47 50 60	syl2anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to ⦋S/u⦌V=Z$$
62	59 61	neeqtrrd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\ne S\right)\right)\to ⦋R/s⦌N\ne ⦋S/u⦌V$$