Theorem 4atexlemnclw

Description: Lemma for 4atexlem7 . (Contributed by NM, 24-Nov-2012)

	Ref	Expression
Hypotheses	4thatlem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
	4thatlem0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	4thatlem0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	4thatlem0.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	4thatlem0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	4thatlem0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	4thatlem0.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	4thatlem0.v	$$⊢V=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
	4thatlem0.c	$$⊢C=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)$$
Assertion	4atexlemnclw	$$⊢\phi \to \neg C\underset{\dot{}}{\le }W$$

Proof

Step	Hyp	Ref	Expression
1		4thatlem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
2		4thatlem0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		4thatlem0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		4thatlem0.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		4thatlem0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		4thatlem0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		4thatlem0.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		4thatlem0.v	$$⊢V=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
9		4thatlem0.c	$$⊢C=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)$$
10	1	4atexlemkl	$$⊢\phi \to K\in \mathrm{Lat}$$
11	1 3 5	4atexlemqtb	$$⊢\phi \to Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
12	1 3 5	4atexlempsb	$$⊢\phi \to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
13		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
14	13 2 4	latmle1	$$⊢\left(K\in \mathrm{Lat}\wedge Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\right)\to \left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
15	10 11 12 14	syl3anc	$$⊢\phi \to \left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
16	9 15	eqbrtrid	$$⊢\phi \to C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
17		simp13r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg Q\underset{\dot{}}{\le }W$$
18	1 17	sylbi	$$⊢\phi \to \neg Q\underset{\dot{}}{\le }W$$
19	1	4atexlemkc	$$⊢\phi \to K\in \mathrm{CvLat}$$
20	1 2 3 4 5 6 7 8	4atexlemv	$$⊢\phi \to V\in A$$
21	1	4atexlemq	$$⊢\phi \to Q\in A$$
22	1	4atexlemt	$$⊢\phi \to T\in A$$
23	1 2 3 4 5 6 7	4atexlemu	$$⊢\phi \to U\in A$$
24	1 2 3 4 5 6 7 8	4atexlemunv	$$⊢\phi \to U\ne V$$
25	1	4atexlemutvt	$$⊢\phi \to U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T$$
26	5 3	cvlsupr6	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(U\in A\wedge V\in A\wedge T\in A\right)\wedge \left(U\ne V\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\to T\ne V$$
27	26	necomd	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(U\in A\wedge V\in A\wedge T\in A\right)\wedge \left(U\ne V\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\to V\ne T$$
28	19 23 20 22 24 25 27	syl132anc	$$⊢\phi \to V\ne T$$
29	2 3 5	cvlatexch2	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(V\in A\wedge Q\in A\wedge T\in A\right)\wedge V\ne T\right)\to \left(V\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\to Q\underset{\dot{}}{\le }\left(V\underset{\dot{}}{\vee }T\right)\right)$$
30	19 20 21 22 28 29	syl131anc	$$⊢\phi \to \left(V\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\to Q\underset{\dot{}}{\le }\left(V\underset{\dot{}}{\vee }T\right)\right)$$
31	1 6	4atexlemwb	$$⊢\phi \to W\in {\mathrm{Base}}_K$$
32	13 2 4	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
33	10 12 31 32	syl3anc	$$⊢\phi \to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
34	8 33	eqbrtrid	$$⊢\phi \to V\underset{\dot{}}{\le }W$$
35	1 2 3 4 5 6 7 8	4atexlemtlw	$$⊢\phi \to T\underset{\dot{}}{\le }W$$
36	13 5	atbase	$$⊢V\in A\to V\in {\mathrm{Base}}_K$$
37	20 36	syl	$$⊢\phi \to V\in {\mathrm{Base}}_K$$
38	13 5	atbase	$$⊢T\in A\to T\in {\mathrm{Base}}_K$$
39	22 38	syl	$$⊢\phi \to T\in {\mathrm{Base}}_K$$
40	13 2 3	latjle12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(V\in {\mathrm{Base}}_K\wedge T\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\right)\to \left(\left(V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }W\right)\leftrightarrow \left(V\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }W\right)$$
41	10 37 39 31 40	syl13anc	$$⊢\phi \to \left(\left(V\underset{\dot{}}{\le }W\wedge T\underset{\dot{}}{\le }W\right)\leftrightarrow \left(V\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }W\right)$$
42	34 35 41	mpbi2and	$$⊢\phi \to \left(V\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }W$$
43	13 5	atbase	$$⊢Q\in A\to Q\in {\mathrm{Base}}_K$$
44	21 43	syl	$$⊢\phi \to Q\in {\mathrm{Base}}_K$$
45	1	4atexlemk	$$⊢\phi \to K\in \mathrm{HL}$$
46	13 3 5	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge V\in A\wedge T\in A\right)\to V\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
47	45 20 22 46	syl3anc	$$⊢\phi \to V\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
48	13 2	lattr	$$⊢\left(K\in \mathrm{Lat}\wedge \left(Q\in {\mathrm{Base}}_K\wedge V\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\right)\to \left(\left(Q\underset{\dot{}}{\le }\left(V\underset{\dot{}}{\vee }T\right)\wedge \left(V\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }W\right)\to Q\underset{\dot{}}{\le }W\right)$$
49	10 44 47 31 48	syl13anc	$$⊢\phi \to \left(\left(Q\underset{\dot{}}{\le }\left(V\underset{\dot{}}{\vee }T\right)\wedge \left(V\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\le }W\right)\to Q\underset{\dot{}}{\le }W\right)$$
50	42 49	mpan2d	$$⊢\phi \to \left(Q\underset{\dot{}}{\le }\left(V\underset{\dot{}}{\vee }T\right)\to Q\underset{\dot{}}{\le }W\right)$$
51	30 50	syld	$$⊢\phi \to \left(V\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\to Q\underset{\dot{}}{\le }W\right)$$
52	18 51	mtod	$$⊢\phi \to \neg V\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
53		nbrne2	$$⊢\left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge \neg V\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\right)\to C\ne V$$
54	16 52 53	syl2anc	$$⊢\phi \to C\ne V$$
55	1	4atexlemw	$$⊢\phi \to W\in H$$
56	45 55	jca	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
57	1	4atexlempw	$$⊢\phi \to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
58	1	4atexlems	$$⊢\phi \to S\in A$$
59	1 2 3 4 5 6 7 8 9	4atexlemc	$$⊢\phi \to C\in A$$
60	1 2 3 5	4atexlempns	$$⊢\phi \to P\ne S$$
61	13 2 4	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\right)\to \left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
62	10 11 12 61	syl3anc	$$⊢\phi \to \left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
63	9 62	eqbrtrid	$$⊢\phi \to C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
64	2 3 4 5 6 8	lhpat3	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge C\in A\right)\wedge \left(P\ne S\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\to \left(\neg C\underset{\dot{}}{\le }W\leftrightarrow C\ne V\right)$$
65	56 57 58 59 60 63 64	syl222anc	$$⊢\phi \to \left(\neg C\underset{\dot{}}{\le }W\leftrightarrow C\ne V\right)$$
66	54 65	mpbird	$$⊢\phi \to \neg C\underset{\dot{}}{\le }W$$