Theorem 4atexlemcnd 34376

Description: Lemma for 4atexlem7 34379. (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 34371	. . 3 ⊢ (𝜑 → 𝑇 ≤ 𝑊)
10		4thatlem0.c	. . . 4 ⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
11	1, 2, 3, 4, 5, 6, 7, 8, 10	4atexlemnclw 34374	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝑊)
12		nbrne2 4603	. . 3 ⊢ ((𝑇 ≤ 𝑊 ∧ ¬ 𝐶 ≤ 𝑊) → 𝑇 ≠ 𝐶)
13	9, 11, 12	syl2anc 691	. 2 ⊢ (𝜑 → 𝑇 ≠ 𝐶)
14	1	4atexlemk 34351	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ HL)
15	1	4atexlemq 34355	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
16	1	4atexlemt 34357	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
17	3, 5	hlatjcom 33672	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
18	14, 15, 16, 17	syl3anc 1318	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
19		simp221 1195	. . . . . . . . . 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 206	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
21	3, 5	hlatjcom 33672	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅))
22	14, 20, 16, 21	syl3anc 1318	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ∨ 𝑇) = (𝑇 ∨ 𝑅))
23	18, 22	oveq12d 6567	. . . . . . 7 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) = ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)))
24	1	4atexlemkc 34362	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ CvLat)
25	1	4atexlemp 34354	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
26	1	4atexlempnq 34359	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
27		simp223 1197	. . . . . . . . . 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 206	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
29	5, 3	cvlsupr6 33652	. . . . . . . . . 10 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))) → 𝑅 ≠ 𝑄)
30	29	necomd 2837	. . . . . . . . 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 1336	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≠ 𝑅)
32	1, 2, 3, 4, 5, 6, 7, 8	4atexlemntlpq 34372	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
33	5, 3	cvlsupr7 33653	. . . . . . . . . . . 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 1336	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄))
35	3, 5	hlatjcom 33672	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
36	14, 15, 20, 35	syl3anc 1318	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
37	34, 36	eqtr4d 2647	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑅))
38	37	breq2d 4595	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ≤ (𝑃 ∨ 𝑄) ↔ 𝑇 ≤ (𝑄 ∨ 𝑅)))
39	32, 38	mtbid 313	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑄 ∨ 𝑅))
40	2, 3, 4, 5	2llnma2 34093	. . . . . . . 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 1336	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∨ 𝑄) ∧ (𝑇 ∨ 𝑅)) = 𝑇)
42	23, 41	eqtr2d 2645	. . . . . 6 ⊢ (𝜑 → 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
43	42	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝑇 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
44	1	4atexlemkl 34361	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Lat)
45	1, 3, 5	4atexlemqtb 34365	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
46	1, 3, 5	4atexlempsb 34364	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
47		eqid 2610	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
48	47, 2, 4	latmle1 16899	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
49	44, 45, 46, 48	syl3anc 1318	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
50	10, 49	syl5eqbr 4618	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
51	50	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ (𝑄 ∨ 𝑇))
52		simpr 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = 𝐷)
53		4thatlem0.d	. . . . . . . . . 10 ⊢ 𝐷 = ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
54	47, 3, 5	hlatjcl 33671	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑅 ∨ 𝑇) ∈ (Base‘𝐾))
55	14, 20, 16, 54	syl3anc 1318	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ∨ 𝑇) ∈ (Base‘𝐾))
56	47, 2, 4	latmle1 16899	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑅 ∨ 𝑇))
57	44, 55, 46, 56	syl3anc 1318	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑅 ∨ 𝑇))
58	53, 57	syl5eqbr 4618	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ≤ (𝑅 ∨ 𝑇))
59	58	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐷 ≤ (𝑅 ∨ 𝑇))
60	52, 59	eqbrtrd 4605	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ (𝑅 ∨ 𝑇))
61	1, 2, 3, 4, 5, 6, 7, 8, 10	4atexlemc 34373	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
62	47, 5	atbase 33594	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ (Base‘𝐾))
63	61, 62	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
64	47, 2, 4	latlem12 16901	. . . . . . . . 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 1320	. . . . . . . 8 ⊢ (𝜑 → ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
66	65	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → ((𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
67	51, 60, 66	mpbi2and 958	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
68		hlatl 33665	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
69	14, 68	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ AtLat)
70	42, 16	eqeltrrd 2689	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ∈ 𝐴)
71	2, 5	atcmp 33616	. . . . . . . 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 1318	. . . . . . 7 ⊢ (𝜑 → (𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
73	72	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐶 ≤ ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)) ↔ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇))))
74	67, 73	mpbid 221	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑅 ∨ 𝑇)))
75	43, 74	eqtr4d 2647	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝑇 = 𝐶)
76	75	ex 449	. . 3 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝑇 = 𝐶))
77	76	necon3d 2803	. 2 ⊢ (𝜑 → (𝑇 ≠ 𝐶 → 𝐶 ≠ 𝐷))
78	13, 77	mpd 15	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Latclat 16868  Atomscatm 33568  AtLatcal 33569  CvLatclc 33570  HLchlt 33655  LHypclh 34288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lhyp 34292

This theorem is referenced by:  4atexlemex4  34377