Theorem cdleme42b

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 6-Mar-2013)

	Ref	Expression
Hypotheses	cdleme41.b	$$⊢B={\mathrm{Base}}_K$$
	cdleme41.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme41.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme41.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme41.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme41.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme41.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme41.d	$$⊢D=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme41.e	$$⊢E=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme41.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme41.i	$$⊢I=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=G\right)\right)$$
	cdleme41.n	$$⊢N=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,D\right)$$
	cdleme41.o	$$⊢O=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right)$$
	cdleme41.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	cdleme42b	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to F\left(X\right)=⦋R/s⦌N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme41.b	$$⊢B={\mathrm{Base}}_K$$
2		cdleme41.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdleme41.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdleme41.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdleme41.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdleme41.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdleme41.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdleme41.d	$$⊢D=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme41.e	$$⊢E=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdleme41.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
11		cdleme41.i	$$⊢I=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=G\right)\right)$$
12		cdleme41.n	$$⊢N=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,D\right)$$
13		cdleme41.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		cdleme41.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	1	fvexi	$$⊢B\in \mathrm{V}$$
16		nfv	$$⊢Ⅎs\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)$$
17		nfcsb1v	$$⊢\underset{\_}{Ⅎ}s⦋R/s⦌N$$
18		nfcv	$$⊢\underset{\_}{Ⅎ}s\underset{\dot{}}{\vee }$$
19		nfcv	$$⊢\underset{\_}{Ⅎ}s\left(X\underset{\dot{}}{\wedge }W\right)$$
20	17 18 19	nfov	$$⊢\underset{\_}{Ⅎ}s\left(⦋R/s⦌N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)$$
21	20	a1i	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to \underset{\_}{Ⅎ}s\left(⦋R/s⦌N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)$$
22		nfvd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to Ⅎs\left(\neg R\underset{\dot{}}{\le }W\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)$$
23		eqid	$$⊢\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
24	13 14 23	cdleme31fv1	$$⊢\left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to F\left(X\right)=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
25	24	3ad2ant2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to F\left(X\right)=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
26		breq1	$$⊢s=R\to \left(s\underset{\dot{}}{\le }W\leftrightarrow R\underset{\dot{}}{\le }W\right)$$
27	26	notbid	$$⊢s=R\to \left(\neg s\underset{\dot{}}{\le }W\leftrightarrow \neg R\underset{\dot{}}{\le }W\right)$$
28		oveq1	$$⊢s=R\to s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)$$
29	28	eqeq1d	$$⊢s=R\to \left(s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\leftrightarrow R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)$$
30	27 29	anbi12d	$$⊢s=R\to \left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\leftrightarrow \left(\neg R\underset{\dot{}}{\le }W\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)$$
31	30	adantl	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\wedge s=R\right)\to \left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\leftrightarrow \left(\neg R\underset{\dot{}}{\le }W\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)$$
32		csbeq1a	$$⊢s=R\to N=⦋R/s⦌N$$
33	32	oveq1d	$$⊢s=R\to N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=⦋R/s⦌N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)$$
34	33	adantl	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\wedge s=R\right)\to N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=⦋R/s⦌N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)$$
35		simp1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to \left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)$$
36		simp2l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to X\in B$$
37	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme32fvcl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge X\in B\right)\to F\left(X\right)\in B$$
38	35 36 37	syl2anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to F\left(X\right)\in B$$
39		simp3ll	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to R\in A$$
40		simp3lr	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to \neg R\underset{\dot{}}{\le }W$$
41		simp3r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X$$
42	40 41	jca	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to \left(\neg R\underset{\dot{}}{\le }W\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)$$
43	16 21 22 25 31 34 38 39 42	riotasv2d	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\wedge B\in \mathrm{V}\right)\to F\left(X\right)=⦋R/s⦌N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)$$
44	15 43	mpan2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge R\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to F\left(X\right)=⦋R/s⦌N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)$$