Theorem 4atexlemcnd 40771

Description: Lemma for 4atexlem7 40774. (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 40766	. . 3 ⊢ (𝜑 → 𝑇 ≤ 𝑊)
10		4thatlem0.c	. . . 4 ⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
11	1, 2, 3, 4, 5, 6, 7, 8, 10	4atexlemnclw 40769	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝑊)
12		nbrne2 5133	. . 3 ⊢ ((𝑇 ≤ 𝑊 ∧ ¬ 𝐶 ≤ 𝑊) → 𝑇 ≠ 𝐶)
13	9, 11, 12	syl2anc 595	. 2 ⊢ (𝜑 → 𝑇 ≠ 𝐶)
14	1	4atexlemk 40746	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ HL)
15	1	4atexlemq 40750	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
16	1	4atexlemt 40752	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
17	3, 5	hlatjcom 40067	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
18	14, 15, 16, 17	syl3anc 1396	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
19		simp221 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))) → 𝑅 ∈ 𝐴)
20	1, 19	sylbi 220	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
21	3, 5	hlatjcom 40067	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅))
22	14, 20, 16, 21	syl3anc 1396	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅))
23	18, 22	oveq12d 7429	. . . . . . 7 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) = ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)))
24	1	4atexlemkc 40757	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CvLat)
25	1	4atexlemp 40749	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
26	1	4atexlempnq 40754	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
27		simp223 1333	. . . . . . . . . 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 220	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
29	5, 3	cvlsupr6 40046	. . . . . . . . . 10 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → 𝑅 ≠ 𝑄)
30	29	necomd 3019	. . . . . . . . 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 1413	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≠ 𝑅)
32	1, 2, 3, 4, 5, 6, 7, 8	4atexlemntlpq 40767	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
33	5, 3	cvlsupr7 40047	. . . . . . . . . . . 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 1413	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄))
35	3, 5	hlatjcom 40067	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
36	14, 15, 20, 35	syl3anc 1396	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
37	34, 36	eqtr4d 2807	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑅))
38	37	breq2d 5123	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ≤ (𝑃 ∨ 𝑄) ↔ 𝑇 ≤ (𝑄 ∨ 𝑅)))
39	32, 38	mtbid 327	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑄 ∨ 𝑅))
40	2, 3, 4, 5	2llnma2 40488	. . . . . . . 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 1413	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)) = 𝑇)
42	23, 41	eqtr2d 2805	. . . . . 6 ⊢ (𝜑 → 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
43	42	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
44	1	4atexlemkl 40756	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Lat)
45	1, 3, 5	4atexlemqtb 40760	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
46	1, 3, 5	4atexlempsb 40759	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
47		eqid 2769	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
48	47, 2, 4	latmle1 18520	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
49	44, 45, 46, 48	syl3anc 1396	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
50	10, 49	eqbrtrid 5148	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
51	50	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ (𝑄 ∨ 𝑇))
52		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = 𝐷)
53		4thatlem0.d	. . . . . . . . . 10 ⊢ 𝐷 = ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
54	47, 3, 5	hlatjcl 40066	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑅 ∨ 𝑇) ∈ (Base‘𝐾))
55	14, 20, 16, 54	syl3anc 1396	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ∨ 𝑇) ∈ (Base‘𝐾))
56	47, 2, 4	latmle1 18520	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑅 ∨ 𝑇))
57	44, 55, 46, 56	syl3anc 1396	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑅 ∨ 𝑇))
58	53, 57	eqbrtrid 5148	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ≤ (𝑅 ∨ 𝑇))
59	58	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐷 ≤ (𝑅 ∨ 𝑇))
60	52, 59	eqbrtrd 5135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ (𝑅 ∨ 𝑇))
61	1, 2, 3, 4, 5, 6, 7, 8, 10	4atexlemc 40768	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
62	47, 5	atbase 39988	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ (Base‘𝐾))
63	61, 62	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
64	47, 2, 4	latlem12 18522	. . . . . . . . 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 1397	. . . . . . . 8 ⊢ (𝜑 → ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
66	65	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
67	51, 60, 66	mpbi2and 724	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
68		hlatl 40059	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
69	14, 68	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ AtLat)
70	42, 16	eqeltrrd 2870	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ∈ 𝐴)
71	2, 5	atcmp 40010	. . . . . . . 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 1396	. . . . . . 7 ⊢ (𝜑 → (𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
73	72	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
74	67, 73	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
75	43, 74	eqtr4d 2807	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝑇 = 𝐶)
76	75	ex 417	. . 3 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝑇 = 𝐶))
77	76	necon3d 2985	. 2 ⊢ (𝜑 → (𝑇 ≠ 𝐶 → 𝐶 ≠ 𝐷))
78	13, 77	mpd 16	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   class class class wbr 5111  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  lecple 17317  joincjn 18367  meetcmee 18368  Latclat 18487  Atomscatm 39962  AtLatcal 39963  CvLatclc 39964  HLchlt 40049  LHypclh 40683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-proset 18350  df-poset 18369  df-plt 18384  df-lub 18400  df-glb 18401  df-join 18402  df-meet 18403  df-p0 18479  df-p1 18480  df-lat 18488  df-clat 18555  df-oposet 39875  df-ol 39877  df-oml 39878  df-covers 39965  df-ats 39966  df-atl 39997  df-cvlat 40021  df-hlat 40050  df-llines 40197  df-lplanes 40198  df-lhyp 40687

This theorem is referenced by:  4atexlemex4  40772