Theorem 4atexlemcnd 36142

Description: Lemma for 4atexlem7 36145. (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 36137	. . 3 ⊢ (𝜑 → 𝑇 ≤ 𝑊)
10		4thatlem0.c	. . . 4 ⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
11	1, 2, 3, 4, 5, 6, 7, 8, 10	4atexlemnclw 36140	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝑊)
12		nbrne2 4895	. . 3 ⊢ ((𝑇 ≤ 𝑊 ∧ ¬ 𝐶 ≤ 𝑊) → 𝑇 ≠ 𝐶)
13	9, 11, 12	syl2anc 579	. 2 ⊢ (𝜑 → 𝑇 ≠ 𝐶)
14	1	4atexlemk 36117	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ HL)
15	1	4atexlemq 36121	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
16	1	4atexlemt 36123	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
17	3, 5	hlatjcom 35438	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
18	14, 15, 16, 17	syl3anc 1494	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
19		simp221 1417	. . . . . . . . . 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 209	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
21	3, 5	hlatjcom 35438	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅))
22	14, 20, 16, 21	syl3anc 1494	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅))
23	18, 22	oveq12d 6928	. . . . . . 7 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) = ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)))
24	1	4atexlemkc 36128	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CvLat)
25	1	4atexlemp 36120	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
26	1	4atexlempnq 36125	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
27		simp223 1419	. . . . . . . . . 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 209	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
29	5, 3	cvlsupr6 35417	. . . . . . . . . 10 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → 𝑅 ≠ 𝑄)
30	29	necomd 3054	. . . . . . . . 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 1511	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≠ 𝑅)
32	1, 2, 3, 4, 5, 6, 7, 8	4atexlemntlpq 36138	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
33	5, 3	cvlsupr7 35418	. . . . . . . . . . . 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 1511	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄))
35	3, 5	hlatjcom 35438	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
36	14, 15, 20, 35	syl3anc 1494	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
37	34, 36	eqtr4d 2864	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑅))
38	37	breq2d 4887	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ≤ (𝑃 ∨ 𝑄) ↔ 𝑇 ≤ (𝑄 ∨ 𝑅)))
39	32, 38	mtbid 316	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑄 ∨ 𝑅))
40	2, 3, 4, 5	2llnma2 35859	. . . . . . . 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 1511	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)) = 𝑇)
42	23, 41	eqtr2d 2862	. . . . . 6 ⊢ (𝜑 → 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
43	42	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
44	1	4atexlemkl 36127	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Lat)
45	1, 3, 5	4atexlemqtb 36131	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
46	1, 3, 5	4atexlempsb 36130	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
47		eqid 2825	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
48	47, 2, 4	latmle1 17436	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
49	44, 45, 46, 48	syl3anc 1494	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
50	10, 49	syl5eqbr 4910	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
51	50	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ (𝑄 ∨ 𝑇))
52		simpr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = 𝐷)
53		4thatlem0.d	. . . . . . . . . 10 ⊢ 𝐷 = ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
54	47, 3, 5	hlatjcl 35437	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑅 ∨ 𝑇) ∈ (Base‘𝐾))
55	14, 20, 16, 54	syl3anc 1494	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ∨ 𝑇) ∈ (Base‘𝐾))
56	47, 2, 4	latmle1 17436	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑅 ∨ 𝑇))
57	44, 55, 46, 56	syl3anc 1494	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑅 ∨ 𝑇))
58	53, 57	syl5eqbr 4910	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ≤ (𝑅 ∨ 𝑇))
59	58	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐷 ≤ (𝑅 ∨ 𝑇))
60	52, 59	eqbrtrd 4897	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ (𝑅 ∨ 𝑇))
61	1, 2, 3, 4, 5, 6, 7, 8, 10	4atexlemc 36139	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
62	47, 5	atbase 35359	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ (Base‘𝐾))
63	61, 62	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
64	47, 2, 4	latlem12 17438	. . . . . . . . 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 1495	. . . . . . . 8 ⊢ (𝜑 → ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
66	65	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
67	51, 60, 66	mpbi2and 703	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
68		hlatl 35430	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
69	14, 68	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ AtLat)
70	42, 16	eqeltrrd 2907	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ∈ 𝐴)
71	2, 5	atcmp 35381	. . . . . . . 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 1494	. . . . . . 7 ⊢ (𝜑 → (𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
73	72	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
74	67, 73	mpbid 224	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
75	43, 74	eqtr4d 2864	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝑇 = 𝐶)
76	75	ex 403	. . 3 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝑇 = 𝐶))
77	76	necon3d 3020	. 2 ⊢ (𝜑 → (𝑇 ≠ 𝐶 → 𝐶 ≠ 𝐷))
78	13, 77	mpd 15	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999   class class class wbr 4875  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  lecple 16319  joincjn 17304  meetcmee 17305  Latclat 17405  Atomscatm 35333  AtLatcal 35334  CvLatclc 35335  HLchlt 35420  LHypclh 36054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-proset 17288  df-poset 17306  df-plt 17318  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-p0 17399  df-p1 17400  df-lat 17406  df-clat 17468  df-oposet 35246  df-ol 35248  df-oml 35249  df-covers 35336  df-ats 35337  df-atl 35368  df-cvlat 35392  df-hlat 35421  df-llines 35568  df-lplanes 35569  df-lhyp 36058

This theorem is referenced by:  4atexlemex4  36143