Theorem dalem21 36824

Description: Lemma for dath 36866. 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 36753	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
3	2	3ad2ant1 1129	. 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 36822	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (LLines‘𝐾))
9	8	3adant2 1127	. 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 36783	. . 3 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾))
13	12	3ad2ant1 1129	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾))
14	1, 4, 5, 6, 10, 11	dalemply 36784	. . . . . . 7 ⊢ (𝜑 → 𝑃 ≤ 𝑌)
15	14	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑃 ≤ 𝑌)
16		dalem21.z	. . . . . . 7 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
17	1, 4, 5, 6, 16	dalemsly 36785	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌)
18	1	dalemkelat 36754	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
19	1, 6	dalempeb 36769	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
20	1, 6	dalemseb 36772	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
21	1, 10	dalemyeb 36779	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
22		eqid 2821	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
23	22, 4, 5	latjle12 17666	. . . . . . . 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 1368	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ≤ 𝑌 ∧ 𝑆 ≤ 𝑌) ↔ (𝑃 ∨ 𝑆) ≤ 𝑌))
25	24	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ((𝑃 ≤ 𝑌 ∧ 𝑆 ≤ 𝑌) ↔ (𝑃 ∨ 𝑆) ≤ 𝑌))
26	15, 17, 25	mpbi2and 710	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → (𝑃 ∨ 𝑆) ≤ 𝑌)
27	26	3adant3 1128	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ≤ 𝑌)
28	7	dalem-ccly 36815	. . . . . . 7 ⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌)
29	28	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ 𝑌)
30	18	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ Lat)
31	7, 6	dalemcceb 36819	. . . . . . . . 9 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
32	31	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
33	7	dalemddea 36814	. . . . . . . . . 10 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
34	22, 6	atbase 36419	. . . . . . . . . 10 ⊢ (𝑑 ∈ 𝐴 → 𝑑 ∈ (Base‘𝐾))
35	33, 34	syl 17	. . . . . . . . 9 ⊢ (𝜓 → 𝑑 ∈ (Base‘𝐾))
36	35	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑑 ∈ (Base‘𝐾))
37	22, 4, 5	latlej1 17664	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ 𝑑 ∈ (Base‘𝐾)) → 𝑐 ≤ (𝑐 ∨ 𝑑))
38	30, 32, 36, 37	syl3anc 1367	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑐 ≤ (𝑐 ∨ 𝑑))
39		eqid 2821	. . . . . . . . . 10 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
40	22, 39	llnbase 36639	. . . . . . . . 9 ⊢ ((𝑐 ∨ 𝑑) ∈ (LLines‘𝐾) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾))
41	8, 40	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾))
42	21	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
43	22, 4	lattr 17660	. . . . . . . 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 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ≤ (𝑐 ∨ 𝑑) ∧ (𝑐 ∨ 𝑑) ≤ 𝑌) → 𝑐 ≤ 𝑌))
45	38, 44	mpand 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ≤ 𝑌 → 𝑐 ≤ 𝑌))
46	29, 45	mtod 200	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ¬ (𝑐 ∨ 𝑑) ≤ 𝑌)
47	46	3adant2 1127	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ (𝑐 ∨ 𝑑) ≤ 𝑌)
48		nbrne2 5078	. . . 4 ⊢ (((𝑃 ∨ 𝑆) ≤ 𝑌 ∧ ¬ (𝑐 ∨ 𝑑) ≤ 𝑌) → (𝑃 ∨ 𝑆) ≠ (𝑐 ∨ 𝑑))
49	27, 47, 48	syl2anc 586	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ≠ (𝑐 ∨ 𝑑))
50	49	necomd 3071	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑑) ≠ (𝑃 ∨ 𝑆))
51		hlatl 36490	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
52	2, 51	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ AtLat)
53	52	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ AtLat)
54	1	dalempea 36756	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
55	1	dalemsea 36759	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
56	22, 5, 6	hlatjcl 36497	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
57	2, 54, 55, 56	syl3anc 1367	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
58	57	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
59		dalem21.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
60	22, 59	latmcl 17656	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑑) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
61	30, 41, 58, 60	syl3anc 1367	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
62	1, 4, 5, 6, 10, 11	dalemcea 36790	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
63	62	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ 𝐴)
64	7	dalemclccjdd 36818	. . . . . 6 ⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑))
65	64	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ (𝑐 ∨ 𝑑))
66	1	dalemclpjs 36764	. . . . . 6 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
67	66	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ (𝑃 ∨ 𝑆))
68	1, 6	dalemceb 36768	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
69	68	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ (Base‘𝐾))
70	22, 4, 59	latlem12 17682	. . . . . 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 1368	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐶 ≤ (𝑐 ∨ 𝑑) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)) ↔ 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆))))
72	65, 67, 71	mpbi2and 710	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)))
73		eqid 2821	. . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾)
74	22, 4, 73, 6	atlen0 36440	. . . 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 1369	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾))
76	75	3adant2 1127	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾))
77	59, 73, 6, 39	2llnmat 36654	. 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 1374	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016   class class class wbr 5058  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  lecple 16566  joincjn 17548  meetcmee 17549  0.cp0 17641  Latclat 17649  Atomscatm 36393  AtLatcal 36394  HLchlt 36480  LLinesclln 36621  LPlanesclpl 36622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17532  df-poset 17550  df-plt 17562  df-lub 17578  df-glb 17579  df-join 17580  df-meet 17581  df-p0 17643  df-lat 17650  df-clat 17712  df-oposet 36306  df-ol 36308  df-oml 36309  df-covers 36396  df-ats 36397  df-atl 36428  df-cvlat 36452  df-hlat 36481  df-llines 36628  df-lplanes 36629

This theorem is referenced by:  dalem22  36825