Theorem cdleme32fva

Description: Part of proof of Lemma D in Crawley p. 113. Value of F at an atom not under W . (Contributed by NM, 2-Mar-2013)

	Ref	Expression
Hypotheses	cdleme32.b	$$⊢B={\mathrm{Base}}_K$$
	cdleme32.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme32.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme32.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme32.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme32.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme32.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme32.c	$$⊢C=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme32.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme32.e	$$⊢E=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme32.i	$$⊢I=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=E\right)\right)$$
	cdleme32.n	$$⊢N=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,C\right)$$
	cdleme32.o	$$⊢O=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right)$$
	cdleme32.f	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),O,x\right)\right)$$
Assertion	cdleme32fva	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to ⦋R/x⦌O=⦋R/s⦌N$$

Proof

Step	Hyp	Ref	Expression
1		cdleme32.b	$$⊢B={\mathrm{Base}}_K$$
2		cdleme32.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdleme32.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdleme32.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdleme32.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdleme32.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdleme32.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdleme32.c	$$⊢C=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme32.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdleme32.e	$$⊢E=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
11		cdleme32.i	$$⊢I=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=E\right)\right)$$
12		cdleme32.n	$$⊢N=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,C\right)$$
13		cdleme32.o	$$⊢O=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right)$$
14		cdleme32.f	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),O,x\right)\right)$$
15		simp2l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to R\in A$$
16	1 5	atbase	$$⊢R\in A\to R\in B$$
17	15 16	syl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to R\in B$$
18		eqid	$$⊢\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\right)\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\right)=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\right)\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\right)$$
19	13 18	cdleme31so	$$⊢R\in B\to ⦋R/x⦌O=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\right)\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\right)$$
20	17 19	syl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to ⦋R/x⦌O=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\right)\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\right)$$
21		simp1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to \left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)$$
22		simp3	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to P\ne Q$$
23		simp2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)$$
24	1 2 3 4 5 6 7 8 9 10 11 12	cdleme32snb	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\right)\to ⦋R/s⦌N\in B$$
25	21 22 23 24	syl12anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to ⦋R/s⦌N\in B$$
26		nfv	$$⊢Ⅎs\neg R\underset{\dot{}}{\le }W$$
27		nfcsb1v	$$⊢\underset{\_}{Ⅎ}s⦋R/s⦌N$$
28	27	nfeq2	$$⊢Ⅎsz=⦋R/s⦌N$$
29	26 28	nfim	$$⊢Ⅎs\left(\neg R\underset{\dot{}}{\le }W\to z=⦋R/s⦌N\right)$$
30		breq1	$$⊢s=R\to \left(s\underset{\dot{}}{\le }W\leftrightarrow R\underset{\dot{}}{\le }W\right)$$
31	30	notbid	$$⊢s=R\to \left(\neg s\underset{\dot{}}{\le }W\leftrightarrow \neg R\underset{\dot{}}{\le }W\right)$$
32		csbeq1a	$$⊢s=R\to N=⦋R/s⦌N$$
33	32	eqeq2d	$$⊢s=R\to \left(z=N\leftrightarrow z=⦋R/s⦌N\right)$$
34	31 33	imbi12d	$$⊢s=R\to \left(\left(\neg s\underset{\dot{}}{\le }W\to z=N\right)\leftrightarrow \left(\neg R\underset{\dot{}}{\le }W\to z=⦋R/s⦌N\right)\right)$$
35	34	ax-gen	$$⊢\forall s\left(s=R\to \left(\left(\neg s\underset{\dot{}}{\le }W\to z=N\right)\leftrightarrow \left(\neg R\underset{\dot{}}{\le }W\to z=⦋R/s⦌N\right)\right)\right)$$
36		ceqsralt	$$⊢\left(Ⅎs\left(\neg R\underset{\dot{}}{\le }W\to z=⦋R/s⦌N\right)\wedge \forall s\left(s=R\to \left(\left(\neg s\underset{\dot{}}{\le }W\to z=N\right)\leftrightarrow \left(\neg R\underset{\dot{}}{\le }W\to z=⦋R/s⦌N\right)\right)\right)\wedge R\in A\right)\to \left(\forall s\in A\left(s=R\to \left(\neg s\underset{\dot{}}{\le }W\to z=N\right)\right)\leftrightarrow \left(\neg R\underset{\dot{}}{\le }W\to z=⦋R/s⦌N\right)\right)$$
37	29 35 36	mp3an12	$$⊢R\in A\to \left(\forall s\in A\left(s=R\to \left(\neg s\underset{\dot{}}{\le }W\to z=N\right)\right)\leftrightarrow \left(\neg R\underset{\dot{}}{\le }W\to z=⦋R/s⦌N\right)\right)$$
38	37	adantr	$$⊢\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\to \left(\forall s\in A\left(s=R\to \left(\neg s\underset{\dot{}}{\le }W\to z=N\right)\right)\leftrightarrow \left(\neg R\underset{\dot{}}{\le }W\to z=⦋R/s⦌N\right)\right)$$
39	38	3ad2ant2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to \left(\forall s\in A\left(s=R\to \left(\neg s\underset{\dot{}}{\le }W\to z=N\right)\right)\leftrightarrow \left(\neg R\underset{\dot{}}{\le }W\to z=⦋R/s⦌N\right)\right)$$
40		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
41		eqid	$$⊢\mathrm{0.}\left(K\right)=\mathrm{0.}\left(K\right)$$
42	2 4 41 5 6	lhpmat	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\to R\underset{\dot{}}{\wedge }W=\mathrm{0.}\left(K\right)$$
43	40 23 42	syl2anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to R\underset{\dot{}}{\wedge }W=\mathrm{0.}\left(K\right)$$
44	43	adantr	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to R\underset{\dot{}}{\wedge }W=\mathrm{0.}\left(K\right)$$
45	44	oveq2d	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=s\underset{\dot{}}{\vee }\mathrm{0.}\left(K\right)$$
46		simp11l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to K\in \mathrm{HL}$$
47	46	adantr	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to K\in \mathrm{HL}$$
48		hlol	$$⊢K\in \mathrm{HL}\to K\in \mathrm{OL}$$
49	47 48	syl	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to K\in \mathrm{OL}$$
50	1 5	atbase	$$⊢s\in A\to s\in B$$
51	50	ad2antrl	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to s\in B$$
52	1 3 41	olj01	$$⊢\left(K\in \mathrm{OL}\wedge s\in B\right)\to s\underset{\dot{}}{\vee }\mathrm{0.}\left(K\right)=s$$
53	49 51 52	syl2anc	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to s\underset{\dot{}}{\vee }\mathrm{0.}\left(K\right)=s$$
54	45 53	eqtrd	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=s$$
55	54	eqeq1d	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to \left(s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\leftrightarrow s=R\right)$$
56	44	oveq2d	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=N\underset{\dot{}}{\vee }\mathrm{0.}\left(K\right)$$
57		simpl11	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
58		simpl12	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
59		simpl13	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
60		simpr	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)$$
61		simpl3	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to P\ne Q$$
62	1 2 3 4 5 6 7 8 9 10 11 12	cdleme27cl	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\right)\to N\in B$$
63	57 58 59 60 61 62	syl122anc	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to N\in B$$
64	1 3 41	olj01	$$⊢\left(K\in \mathrm{OL}\wedge N\in B\right)\to N\underset{\dot{}}{\vee }\mathrm{0.}\left(K\right)=N$$
65	49 63 64	syl2anc	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to N\underset{\dot{}}{\vee }\mathrm{0.}\left(K\right)=N$$
66	56 65	eqtrd	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=N$$
67	66	eqeq2d	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to \left(z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\leftrightarrow z=N\right)$$
68	55 67	imbi12d	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\right)\to \left(\left(s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow \left(s=R\to z=N\right)\right)$$
69	68	expr	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge s\in A\right)\to \left(\neg s\underset{\dot{}}{\le }W\to \left(\left(s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow \left(s=R\to z=N\right)\right)\right)$$
70	69	pm5.74d	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge s\in A\right)\to \left(\left(\neg s\underset{\dot{}}{\le }W\to \left(s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\right)\leftrightarrow \left(\neg s\underset{\dot{}}{\le }W\to \left(s=R\to z=N\right)\right)\right)$$
71		impexp	$$⊢\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\right)\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow \left(\neg s\underset{\dot{}}{\le }W\to \left(s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\right)$$
72		bi2.04	$$⊢\left(s=R\to \left(\neg s\underset{\dot{}}{\le }W\to z=N\right)\right)\leftrightarrow \left(\neg s\underset{\dot{}}{\le }W\to \left(s=R\to z=N\right)\right)$$
73	70 71 72	3bitr4g	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge s\in A\right)\to \left(\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\right)\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow \left(s=R\to \left(\neg s\underset{\dot{}}{\le }W\to z=N\right)\right)\right)$$
74	73	ralbidva	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to \left(\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\right)\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow \forall s\in A\left(s=R\to \left(\neg s\underset{\dot{}}{\le }W\to z=N\right)\right)\right)$$
75		simp2r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to \neg R\underset{\dot{}}{\le }W$$
76		biimt	$$⊢\neg R\underset{\dot{}}{\le }W\to \left(z=⦋R/s⦌N\leftrightarrow \left(\neg R\underset{\dot{}}{\le }W\to z=⦋R/s⦌N\right)\right)$$
77	75 76	syl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to \left(z=⦋R/s⦌N\leftrightarrow \left(\neg R\underset{\dot{}}{\le }W\to z=⦋R/s⦌N\right)\right)$$
78	39 74 77	3bitr4d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to \left(\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\right)\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow z=⦋R/s⦌N\right)$$
79	78	adantr	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\wedge z\in B\right)\to \left(\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\right)\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow z=⦋R/s⦌N\right)$$
80	25 79	riota5	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to \left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)=R\right)\to z=N\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\wedge }W\right)\right)\right)=⦋R/s⦌N$$
81	20 80	eqtrd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge P\ne Q\right)\to ⦋R/x⦌O=⦋R/s⦌N$$