Theorem dalem21 40528

Description: Lemma for dath 40570. 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 40457	. . 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 40526	. . 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 40487	. . 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 40488	. . . . . . 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 40489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌)
18	1	dalemkelat 40458	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
19	1, 6	dalempeb 40473	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
20	1, 6	dalemseb 40476	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
21	1, 10	dalemyeb 40483	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
22		eqid 2765	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
23	22, 4, 5	latjle12 18530	. . . . . . . 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 1399	. . . . . . 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 40519	. . . . . . 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 40523	. . . . . . . . 9 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
32	31	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
33	7	dalemddea 40518	. . . . . . . . . 10 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
34	22, 6	atbase 40123	. . . . . . . . . 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 18528	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ 𝑑 ∈ (Base‘𝐾)) → 𝑐 ≤ (𝑐 ∨ 𝑑))
38	30, 32, 36, 37	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑐 ≤ (𝑐 ∨ 𝑑))
39		eqid 2765	. . . . . . . . . 10 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
40	22, 39	llnbase 40343	. . . . . . . . 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 18524	. . . . . . . 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 1399	. . . . . . 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 5133	. . . 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 3015	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑑) ≠ (𝑃 ∨ 𝑆))
51		hlatl 40194	. . . . . 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 40460	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
55	1	dalemsea 40463	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
56	22, 5, 6	hlatjcl 40201	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
57	2, 54, 55, 56	syl3anc 1398	. . . . . 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 18520	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑑) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
61	30, 41, 58, 60	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∨ 𝑑) ∧ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
62	1, 4, 5, 6, 10, 11	dalemcea 40494	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
63	62	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ 𝐴)
64	7	dalemclccjdd 40522	. . . . . 6 ⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑))
65	64	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ (𝑐 ∨ 𝑑))
66	1	dalemclpjs 40468	. . . . . 6 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
67	66	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ (𝑃 ∨ 𝑆))
68	1, 6	dalemceb 40472	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
69	68	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ (Base‘𝐾))
70	22, 4, 59	latlem12 18546	. . . . . 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 1399	. . . . 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 2765	. . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾)
74	22, 4, 73, 6	atlen0 40144	. . . 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 1400	. . 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 40358	. 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 1405	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 2146   ≠ wne 2960   class class class wbr 5111  ‘cfv 6540  (class class class)co 7419  Basecbs 17293  lecple 17341  joincjn 18391  meetcmee 18392  0.cp0 18501  Latclat 18511  Atomscatm 40097  AtLatcal 40098  HLchlt 40184  LLinesclln 40325  LPlanesclpl 40326

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-proset 18374  df-poset 18393  df-plt 18408  df-lub 18424  df-glb 18425  df-join 18426  df-meet 18427  df-p0 18503  df-lat 18512  df-clat 18579  df-oposet 40010  df-ol 40012  df-oml 40013  df-covers 40100  df-ats 40101  df-atl 40132  df-cvlat 40156  df-hlat 40185  df-llines 40332  df-lplanes 40333

This theorem is used by:  dalem22  40529