Theorem dalem21 39696

Description: Lemma for dath 39738. 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 39625	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
3	2	3ad2ant1 1134	. 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 39694	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (LLines‘𝐾))
9	8	3adant2 1132	. 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 39655	. . 3 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾))
13	12	3ad2ant1 1134	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾))
14	1, 4, 5, 6, 10, 11	dalemply 39656	. . . . . . 7 ⊢ (𝜑 → 𝑃 ≤ 𝑌)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑃 ≤ 𝑌)
16		dalem21.z	. . . . . . 7 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
17	1, 4, 5, 6, 16	dalemsly 39657	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌)
18	1	dalemkelat 39626	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
19	1, 6	dalempeb 39641	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
20	1, 6	dalemseb 39644	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
21	1, 10	dalemyeb 39651	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
22		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
23	22, 4, 5	latjle12 18495	. . . . . . . 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 1374	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ≤ 𝑌 ∧ 𝑆 ≤ 𝑌) ↔ (𝑃 ∨ 𝑆) ≤ 𝑌))
25	24	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ((𝑃 ≤ 𝑌 ∧ 𝑆 ≤ 𝑌) ↔ (𝑃 ∨ 𝑆) ≤ 𝑌))
26	15, 17, 25	mpbi2and 712	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → (𝑃 ∨ 𝑆) ≤ 𝑌)
27	26	3adant3 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ≤ 𝑌)
28	7	dalem-ccly 39687	. . . . . . 7 ⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌)
29	28	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ 𝑌)
30	18	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ Lat)
31	7, 6	dalemcceb 39691	. . . . . . . . 9 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
32	31	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
33	7	dalemddea 39686	. . . . . . . . . 10 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
34	22, 6	atbase 39290	. . . . . . . . . 10 ⊢ (𝑑 ∈ 𝐴 → 𝑑 ∈ (Base‘𝐾))
35	33, 34	syl 17	. . . . . . . . 9 ⊢ (𝜓 → 𝑑 ∈ (Base‘𝐾))
36	35	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑑 ∈ (Base‘𝐾))
37	22, 4, 5	latlej1 18493	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ 𝑑 ∈ (Base‘𝐾)) → 𝑐 ≤ (𝑐 ∨ 𝑑))
38	30, 32, 36, 37	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑐 ≤ (𝑐 ∨ 𝑑))
39		eqid 2737	. . . . . . . . . 10 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
40	22, 39	llnbase 39511	. . . . . . . . 9 ⊢ ((𝑐 ∨ 𝑑) ∈ (LLines‘𝐾) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾))
41	8, 40	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾))
42	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
43	22, 4	lattr 18489	. . . . . . . 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 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ≤ (𝑐 ∨ 𝑑) ∧ (𝑐 ∨ 𝑑) ≤ 𝑌) → 𝑐 ≤ 𝑌))
45	38, 44	mpand 695	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ≤ 𝑌 → 𝑐 ≤ 𝑌))
46	29, 45	mtod 198	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ¬ (𝑐 ∨ 𝑑) ≤ 𝑌)
47	46	3adant2 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ (𝑐 ∨ 𝑑) ≤ 𝑌)
48		nbrne2 5163	. . . 4 ⊢ (((𝑃 ∨ 𝑆) ≤ 𝑌 ∧ ¬ (𝑐 ∨ 𝑑) ≤ 𝑌) → (𝑃 ∨ 𝑆) ≠ (𝑐 ∨ 𝑑))
49	27, 47, 48	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑆) ≠ (𝑐 ∨ 𝑑))
50	49	necomd 2996	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑑) ≠ (𝑃 ∨ 𝑆))
51		hlatl 39361	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
52	2, 51	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ AtLat)
53	52	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ AtLat)
54	1	dalempea 39628	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
55	1	dalemsea 39631	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
56	22, 5, 6	hlatjcl 39368	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
57	2, 54, 55, 56	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
58	57	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
59		dalem21.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
60	22, 59	latmcl 18485	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑑) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
61	30, 41, 58, 60	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
62	1, 4, 5, 6, 10, 11	dalemcea 39662	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
63	62	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ 𝐴)
64	7	dalemclccjdd 39690	. . . . . 6 ⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑))
65	64	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ (𝑐 ∨ 𝑑))
66	1	dalemclpjs 39636	. . . . . 6 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
67	66	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ (𝑃 ∨ 𝑆))
68	1, 6	dalemceb 39640	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
69	68	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ (Base‘𝐾))
70	22, 4, 59	latlem12 18511	. . . . . 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 1374	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐶 ≤ (𝑐 ∨ 𝑑) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)) ↔ 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆))))
72	65, 67, 71	mpbi2and 712	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)))
73		eqid 2737	. . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾)
74	22, 4, 73, 6	atlen0 39311	. . . 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 1375	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾))
76	75	3adant2 1132	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ≠ (0.‘𝐾))
77	59, 73, 6, 39	2llnmat 39526	. 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 1380	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  0.cp0 18468  Latclat 18476  Atomscatm 39264  AtLatcal 39265  HLchlt 39351  LLinesclln 39493  LPlanesclpl 39494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501

This theorem is referenced by:  dalem22  39697