Theorem 4atexlemcnd 40701

Description: Lemma for 4atexlem7 40704. (Contributed by NM, 24-Nov-2012.)

Hypotheses

Ref	Expression
4thatlem.ph	⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))))
4thatlem0.l	⊢ ≤ = (le‘𝐾)
4thatlem0.j	⊢ ∨ = (join‘𝐾)
4thatlem0.m	⊢ ∧ = (meet‘𝐾)
4thatlem0.a	⊢ 𝐴 = (Atoms‘𝐾)
4thatlem0.h	⊢ 𝐻 = (LHyp‘𝐾)
4thatlem0.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
4thatlem0.v	⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
4thatlem0.c	⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
4thatlem0.d	⊢ 𝐷 = ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))

Assertion

Ref	Expression
4atexlemcnd	⊢ (𝜑 → 𝐶 ≠ 𝐷)

Proof of Theorem 4atexlemcnd

Step	Hyp	Ref	Expression
1		4thatlem.ph	. . . 4 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))))
2		4thatlem0.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		4thatlem0.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		4thatlem0.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
5		4thatlem0.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6		4thatlem0.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
7		4thatlem0.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8		4thatlem0.v	. . . 4 ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
9	1, 2, 3, 4, 5, 6, 7, 8	4atexlemtlw 40696	. . 3 ⊢ (𝜑 → 𝑇 ≤ 𝑊)
10		4thatlem0.c	. . . 4 ⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
11	1, 2, 3, 4, 5, 6, 7, 8, 10	4atexlemnclw 40699	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝑊)
12		nbrne2 5122	. . 3 ⊢ ((𝑇 ≤ 𝑊 ∧ ¬ 𝐶 ≤ 𝑊) → 𝑇 ≠ 𝐶)
13	9, 11, 12	syl2anc 593	. 2 ⊢ (𝜑 → 𝑇 ≠ 𝐶)
14	1	4atexlemk 40676	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ HL)
15	1	4atexlemq 40680	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
16	1	4atexlemt 40682	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
17	3, 5	hlatjcom 39997	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
18	14, 15, 16, 17	syl3anc 1392	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
19		simp221 1329	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ∈ 𝐴)
20	1, 19	sylbi 219	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
21	3, 5	hlatjcom 39997	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅))
22	14, 20, 16, 21	syl3anc 1392	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅))
23	18, 22	oveq12d 7416	. . . . . . 7 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) = ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)))
24	1	4atexlemkc 40687	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CvLat)
25	1	4atexlemp 40679	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
26	1	4atexlempnq 40684	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
27		simp223 1331	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
28	1, 27	sylbi 219	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
29	5, 3	cvlsupr6 39976	. . . . . . . . . 10 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → 𝑅 ≠ 𝑄)
30	29	necomd 3014	. . . . . . . . 9 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → 𝑄 ≠ 𝑅)
31	24, 25, 15, 20, 26, 28, 30	syl132anc 1409	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≠ 𝑅)
32	1, 2, 3, 4, 5, 6, 7, 8	4atexlemntlpq 40697	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
33	5, 3	cvlsupr7 39977	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄))
34	24, 25, 15, 20, 26, 28, 33	syl132anc 1409	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄))
35	3, 5	hlatjcom 39997	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
36	14, 15, 20, 35	syl3anc 1392	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
37	34, 36	eqtr4d 2802	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑅))
38	37	breq2d 5114	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ≤ (𝑃 ∨ 𝑄) ↔ 𝑇 ≤ (𝑄 ∨ 𝑅)))
39	32, 38	mtbid 326	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑄 ∨ 𝑅))
40	2, 3, 4, 5	2llnma2 40418	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ (𝑄 ∨ 𝑅))) → ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)) = 𝑇)
41	14, 15, 20, 16, 31, 39, 40	syl132anc 1409	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)) = 𝑇)
42	23, 41	eqtr2d 2800	. . . . . 6 ⊢ (𝜑 → 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
43	42	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
44	1	4atexlemkl 40686	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Lat)
45	1, 3, 5	4atexlemqtb 40690	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
46	1, 3, 5	4atexlempsb 40689	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
47		eqid 2764	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
48	47, 2, 4	latmle1 18498	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
49	44, 45, 46, 48	syl3anc 1392	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
50	10, 49	eqbrtrid 5137	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
51	50	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ (𝑄 ∨ 𝑇))
52		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = 𝐷)
53		4thatlem0.d	. . . . . . . . . 10 ⊢ 𝐷 = ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
54	47, 3, 5	hlatjcl 39996	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑅 ∨ 𝑇) ∈ (Base‘𝐾))
55	14, 20, 16, 54	syl3anc 1392	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ∨ 𝑇) ∈ (Base‘𝐾))
56	47, 2, 4	latmle1 18498	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑅 ∨ 𝑇))
57	44, 55, 46, 56	syl3anc 1392	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑅 ∨ 𝑇))
58	53, 57	eqbrtrid 5137	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ≤ (𝑅 ∨ 𝑇))
59	58	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐷 ≤ (𝑅 ∨ 𝑇))
60	52, 59	eqbrtrd 5124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ (𝑅 ∨ 𝑇))
61	1, 2, 3, 4, 5, 6, 7, 8, 10	4atexlemc 40698	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
62	47, 5	atbase 39918	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ (Base‘𝐾))
63	61, 62	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
64	47, 2, 4	latlem12 18500	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑅 ∨ 𝑇) ∈ (Base‘𝐾))) → ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
65	44, 63, 45, 55, 64	syl13anc 1393	. . . . . . . 8 ⊢ (𝜑 → ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
66	65	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
67	51, 60, 66	mpbi2and 722	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
68		hlatl 39989	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
69	14, 68	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ AtLat)
70	42, 16	eqeltrrd 2865	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ∈ 𝐴)
71	2, 5	atcmp 39940	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝐶 ∈ 𝐴 ∧ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ∈ 𝐴) → (𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
72	69, 61, 70, 71	syl3anc 1392	. . . . . . 7 ⊢ (𝜑 → (𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
73	72	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
74	67, 73	mpbid 234	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
75	43, 74	eqtr4d 2802	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝑇 = 𝐶)
76	75	ex 416	. . 3 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝑇 = 𝐶))
77	76	necon3d 2980	. 2 ⊢ (𝜑 → (𝑇 ≠ 𝐶 → 𝐶 ≠ 𝐷))
78	13, 77	mpd 15	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ≠ wne 2959   class class class wbr 5102  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  lecple 17295  joincjn 18345  meetcmee 18346  Latclat 18465  Atomscatm 39892  AtLatcal 39893  CvLatclc 39894  HLchlt 39979  LHypclh 40613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-proset 18328  df-poset 18347  df-plt 18362  df-lub 18378  df-glb 18379  df-join 18380  df-meet 18381  df-p0 18457  df-p1 18458  df-lat 18466  df-clat 18533  df-oposet 39805  df-ol 39807  df-oml 39808  df-covers 39895  df-ats 39896  df-atl 39927  df-cvlat 39951  df-hlat 39980  df-llines 40127  df-lplanes 40128  df-lhyp 40617

This theorem is referenced by:  4atexlemex4  40702