Theorem 4atexlemc

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	4atexlemc	$$⊢\phi \to C\in A$$

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 4	latmcom	$$⊢\left(K\in \mathrm{Lat}\wedge Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\right)\to \left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)$$
15	10 11 12 14	syl3anc	$$⊢\phi \to \left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)$$
16	9 15	eqtrid	$$⊢\phi \to C=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)$$
17	1	4atexlemk	$$⊢\phi \to K\in \mathrm{HL}$$
18	1	4atexlemp	$$⊢\phi \to P\in A$$
19	1	4atexlems	$$⊢\phi \to S\in A$$
20	1	4atexlemq	$$⊢\phi \to Q\in A$$
21	1	4atexlemt	$$⊢\phi \to T\in A$$
22	1 2 3 5	4atexlempns	$$⊢\phi \to P\ne S$$
23	1 2 3 4 5 6 7 8	4atexlemntlpq	$$⊢\phi \to \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
24	2 3 5	atnlej2	$$⊢\left(K\in \mathrm{HL}\wedge \left(T\in A\wedge P\in A\wedge Q\in A\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to T\ne Q$$
25	24	necomd	$$⊢\left(K\in \mathrm{HL}\wedge \left(T\in A\wedge P\in A\wedge Q\in A\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to Q\ne T$$
26	17 21 18 20 23 25	syl131anc	$$⊢\phi \to Q\ne T$$
27	1	4atexlempnq	$$⊢\phi \to P\ne Q$$
28	1	4atexlemnslpq	$$⊢\phi \to \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
29	2 3 5	4atlem0ae	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge S\in A\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
30	17 18 20 19 27 28 29	syl132anc	$$⊢\phi \to \neg Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
31	13 5	atbase	$$⊢T\in A\to T\in {\mathrm{Base}}_K$$
32	21 31	syl	$$⊢\phi \to T\in {\mathrm{Base}}_K$$
33	1 2 3 4 5 6 7	4atexlemu	$$⊢\phi \to U\in A$$
34	1 2 3 4 5 6 7 8	4atexlemv	$$⊢\phi \to V\in A$$
35	13 3 5	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge U\in A\wedge V\in A\right)\to U\underset{\dot{}}{\vee }V\in {\mathrm{Base}}_K$$
36	17 33 34 35	syl3anc	$$⊢\phi \to U\underset{\dot{}}{\vee }V\in {\mathrm{Base}}_K$$
37	13 5	atbase	$$⊢Q\in A\to Q\in {\mathrm{Base}}_K$$
38	20 37	syl	$$⊢\phi \to Q\in {\mathrm{Base}}_K$$
39	13 3	latjcl	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\wedge Q\in {\mathrm{Base}}_K\right)\to \left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
40	10 12 38 39	syl3anc	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
41	1	4atexlemkc	$$⊢\phi \to K\in \mathrm{CvLat}$$
42	1 2 3 4 5 6 7 8	4atexlemunv	$$⊢\phi \to U\ne V$$
43	1	4atexlemutvt	$$⊢\phi \to U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T$$
44	5 2 3	cvlsupr4	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(U\in A\wedge V\in A\wedge T\in A\right)\wedge \left(U\ne V\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\to T\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }V\right)$$
45	41 33 34 21 42 43 44	syl132anc	$$⊢\phi \to T\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }V\right)$$
46	13 3 5	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
47	17 18 20 46	syl3anc	$$⊢\phi \to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
48	1 6	4atexlemwb	$$⊢\phi \to W\in {\mathrm{Base}}_K$$
49	13 2 4	latmle1	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
50	10 47 48 49	syl3anc	$$⊢\phi \to \left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
51	7 50	eqbrtrid	$$⊢\phi \to U\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
52	13 2 4	latmle1	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
53	10 12 48 52	syl3anc	$$⊢\phi \to \left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
54	8 53	eqbrtrid	$$⊢\phi \to V\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
55	13 5	atbase	$$⊢U\in A\to U\in {\mathrm{Base}}_K$$
56	33 55	syl	$$⊢\phi \to U\in {\mathrm{Base}}_K$$
57	13 5	atbase	$$⊢V\in A\to V\in {\mathrm{Base}}_K$$
58	34 57	syl	$$⊢\phi \to V\in {\mathrm{Base}}_K$$
59	13 2 3	latjlej12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(U\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\right)\wedge \left(V\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\right)\right)\to \left(\left(U\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge V\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\to \left(U\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)$$
60	10 56 47 58 12 59	syl122anc	$$⊢\phi \to \left(\left(U\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge V\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\to \left(U\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)$$
61	51 54 60	mp2and	$$⊢\phi \to \left(U\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)\right)$$
62	3 5	hlatjass	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge S\in A\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }S=P\underset{\dot{}}{\vee }\left(Q\underset{\dot{}}{\vee }S\right)$$
63	17 18 20 19 62	syl13anc	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }S=P\underset{\dot{}}{\vee }\left(Q\underset{\dot{}}{\vee }S\right)$$
64	13 5	atbase	$$⊢P\in A\to P\in {\mathrm{Base}}_K$$
65	18 64	syl	$$⊢\phi \to P\in {\mathrm{Base}}_K$$
66	13 5	atbase	$$⊢S\in A\to S\in {\mathrm{Base}}_K$$
67	19 66	syl	$$⊢\phi \to S\in {\mathrm{Base}}_K$$
68	13 3	latj32	$$⊢\left(K\in \mathrm{Lat}\wedge \left(P\in {\mathrm{Base}}_K\wedge Q\in {\mathrm{Base}}_K\wedge S\in {\mathrm{Base}}_K\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }S=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }Q$$
69	10 65 38 67 68	syl13anc	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }S=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }Q$$
70	13 3	latjjdi	$$⊢\left(K\in \mathrm{Lat}\wedge \left(P\in {\mathrm{Base}}_K\wedge Q\in {\mathrm{Base}}_K\wedge S\in {\mathrm{Base}}_K\right)\right)\to P\underset{\dot{}}{\vee }\left(Q\underset{\dot{}}{\vee }S\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)$$
71	10 65 38 67 70	syl13anc	$$⊢\phi \to P\underset{\dot{}}{\vee }\left(Q\underset{\dot{}}{\vee }S\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)$$
72	63 69 71	3eqtr3rd	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }Q$$
73	61 72	breqtrd	$$⊢\phi \to \left(U\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }Q\right)$$
74	13 2 10 32 36 40 45 73	lattrd	$$⊢\phi \to T\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }Q\right)$$
75	2 3 4 5	2atmat	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge S\in A\right)\wedge \left(Q\in A\wedge T\in A\wedge P\ne S\right)\wedge \left(Q\ne T\wedge \neg Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge T\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\in A$$
76	17 18 19 20 21 22 26 30 74 75	syl333anc	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }T\right)\in A$$
77	16 76	eqeltrd	$$⊢\phi \to C\in A$$