Theorem dalem21 41017

Description: Lemma for dath 41059. Show that lines 𝑐𝑑 and 𝑃𝑆 intersect at an atom. (Contributed by NM, 2-Aug-2012.)

Hypotheses

Ref	Expression
dalem.ph	⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))
dalem.l	⊢ ≤ = (le‘𝐾)
dalem.j	⊢ ∨ = (join‘𝐾)
dalem.a	⊢ 𝐴 = (Atoms‘𝐾)
dalem.ps	⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
dalem21.m	⊢ ∧ = (meet‘𝐾)
dalem21.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem21.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem21.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)

Assertion

Ref	Expression
dalem21	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ 𝐴)

Proof of Theorem dalem21

Step	Hyp	Ref	Expression
1		dalem.ph	. . . 4 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))
2	1	dalemkehl 40946	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
3	2	3ad2ant1 1151	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
4		dalem.l	. . . 4 ⊢ ≤ = (le‘𝐾)
5		dalem.j	. . . 4 ⊢ ∨ = (join‘𝐾)
6		dalem.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
7		dalem.ps	. . . 4 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
8	1, 4, 5, 6, 7	dalemcjden 41015	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (LLines‘𝐾))
9	8	3adant2 1149	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (LLines‘𝐾))
10		dalem21.o	. . . 4 ⊢ 𝑂 = (LPlanes‘𝐾)
11		dalem21.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
12	1, 4, 5, 6, 10, 11	dalempjsen 40976	. . 3 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾))
13	12	3ad2ant1 1151	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾))
14	1, 4, 5, 6, 10, 11	dalemply 40977	. . . . . . 7 ⊢ (𝜑 → 𝑃 ≤ 𝑌)
15	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑃 ≤ 𝑌)
16		dalem21.z	. . . . . . 7 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
17	1, 4, 5, 6, 16	dalemsly 40978	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌)
18	1	dalemkelat 40947	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
19	1, 6	dalempeb 40962	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
20	1, 6	dalemseb 40965	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
21	1, 10	dalemyeb 40972	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
22		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
23	22, 4, 5	latjle12 18674	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑃 ≤ 𝑌 ∧ 𝑆 ≤ 𝑌) ↔ (𝑃 ∨ 𝑆) ≤ 𝑌))
24	18, 19, 20, 21, 23	syl13anc 1400	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ≤ 𝑌 ∧ 𝑆 ≤ 𝑌) ↔ (𝑃 ∨ 𝑆) ≤ 𝑌))
25	24	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ((𝑃 ≤ 𝑌 ∧ 𝑆 ≤ 𝑌) ↔ (𝑃 ∨ 𝑆) ≤ 𝑌))
26	15, 17, 25	mpbi2and 725	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → (𝑃 ∨ 𝑆) ≤ 𝑌)
27	26	3adant3 1150	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ≤ 𝑌)
28	7	dalem-ccly 41008	. . . . . . 7 ⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌)
29	28	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ 𝑌)
30	18	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ Lat)
31	7, 6	dalemcceb 41012	. . . . . . . . 9 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
32	31	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
33	7	dalemddea 41007	. . . . . . . . . 10 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
34	22, 6	atbase 40612	. . . . . . . . . 10 ⊢ (𝑑 ∈ 𝐴 → 𝑑 ∈ (Base‘𝐾))
35	33, 34	syl 18	. . . . . . . . 9 ⊢ (𝜓 → 𝑑 ∈ (Base‘𝐾))
36	35	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑑 ∈ (Base‘𝐾))
37	22, 4, 5	latlej1 18672	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ 𝑑 ∈ (Base‘𝐾)) → 𝑐 ≤ (𝑐 ∨ 𝑑))
38	30, 32, 36, 37	syl3anc 1399	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑐 ≤ (𝑐 ∨ 𝑑))
39		eqid 2763	. . . . . . . . . 10 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
40	22, 39	llnbase 40832	. . . . . . . . 9 ⊢ ((𝑐 ∨ 𝑑) ∈ (LLines‘𝐾) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾))
41	8, 40	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾))
42	21	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
43	22, 4	lattr 18668	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ (𝑐 ∨ 𝑑) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑐 ≤ (𝑐 ∨ 𝑑) ∧ (𝑐 ∨ 𝑑) ≤ 𝑌) → 𝑐 ≤ 𝑌))
44	30, 32, 41, 42, 43	syl13anc 1400	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ≤ (𝑐 ∨ 𝑑) ∧ (𝑐 ∨ 𝑑) ≤ 𝑌) → 𝑐 ≤ 𝑌))
45	38, 44	mpand 708	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ≤ 𝑌 → 𝑐 ≤ 𝑌))
46	29, 45	mtod 201	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ¬ (𝑐 ∨ 𝑑) ≤ 𝑌)
47	46	3adant2 1149	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ (𝑐 ∨ 𝑑) ≤ 𝑌)
48		nbrne2 5131	. . . 4 ⊢ (((𝑃 ∨ 𝑆) ≤ 𝑌 ∧ ¬ (𝑐 ∨ 𝑑) ≤ 𝑌) → (𝑃 ∨ 𝑆) ≠ (𝑐 ∨ 𝑑))
49	27, 47, 48	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ≠ (𝑐 ∨ 𝑑))
50	49	necomd 3014	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑑) ≠ (𝑃 ∨ 𝑆))
51		hlatl 40683	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
52	2, 51	syl 18	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ AtLat)
53	52	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ AtLat)
54	1	dalempea 40949	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
55	1	dalemsea 40952	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
56	22, 5, 6	hlatjcl 40690	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
57	2, 54, 55, 56	syl3anc 1399	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
58	57	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
59		dalem21.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
60	22, 59	latmcl 18664	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑑) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
61	30, 41, 58, 60	syl3anc 1399	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
62	1, 4, 5, 6, 10, 11	dalemcea 40983	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
63	62	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ 𝐴)
64	7	dalemclccjdd 41011	. . . . . 6 ⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑))
65	64	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ (𝑐 ∨ 𝑑))
66	1	dalemclpjs 40957	. . . . . 6 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
67	66	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ (𝑃 ∨ 𝑆))
68	1, 6	dalemceb 40961	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
69	68	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ (Base‘𝐾))
70	22, 4, 59	latlem12 18690	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑐 ∨ 𝑑) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))) → ((𝐶 ≤ (𝑐 ∨ 𝑑) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)) ↔ 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆))))
71	30, 69, 41, 58, 70	syl13anc 1400	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐶 ≤ (𝑐 ∨ 𝑑) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)) ↔ 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆))))
72	65, 67, 71	mpbi2and 725	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)))
73		eqid 2763	. . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾)
74	22, 4, 73, 6	atlen0 40633	. . . 4 ⊢ (((𝐾 ∈ AtLat ∧ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾) ∧ 𝐶 ∈ 𝐴) ∧ 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆))) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾))
75	53, 61, 63, 72, 74	syl31anc 1401	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾))
76	75	3adant2 1149	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾))
77	59, 73, 6, 39	2llnmat 40847	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑐 ∨ 𝑑) ∈ (LLines‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾)) ∧ ((𝑐 ∨ 𝑑) ≠ (𝑃 ∨ 𝑆) ∧ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾))) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ 𝐴)
78	3, 9, 13, 50, 76, 77	syl32anc 1406	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  lecple 17479  joincjn 18535  meetcmee 18536  0.cp0 18645  Latclat 18655  Atomscatm 40586  AtLatcal 40587  HLchlt 40673  LLinesclln 40814  LPlanesclpl 40815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-proset 18518  df-poset 18537  df-plt 18552  df-lub 18568  df-glb 18569  df-join 18570  df-meet 18571  df-p0 18647  df-lat 18656  df-clat 18723  df-oposet 40499  df-ol 40501  df-oml 40502  df-covers 40589  df-ats 40590  df-atl 40621  df-cvlat 40645  df-hlat 40674  df-llines 40821  df-lplanes 40822

This theorem is used by:  dalem22  41018