Theorem dalem52 40927

Description: Lemma for dath 40939. Lines 𝐺𝐻 and 𝑃𝑄 intersect at an atom. (Contributed by NM, 8-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))))
dalem44.m	⊢ ∧ = (meet‘𝐾)
dalem44.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem44.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem44.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem44.g	⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
dalem44.h	⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
dalem44.i	⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))

Assertion

Ref	Expression
dalem52	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)

Proof of Theorem dalem52

Step	Hyp	Ref	Expression
1		dalem.ph	. . . . 5 ⊢ (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 40826	. . . 4 ⊢ (𝜑 → 𝐾 ∈ HL)
3	2	3ad2ant1 1151	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
4		dalem.ps	. . . . 5 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
5		dalem.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	dalemcceb 40892	. . . 4 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
7	6	3ad2ant3 1153	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
8	3, 7	jca 521	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)))
9		dalem.l	. . . 4 ⊢ ≤ = (le‘𝐾)
10		dalem.j	. . . 4 ⊢ ∨ = (join‘𝐾)
11		dalem44.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
12		dalem44.o	. . . 4 ⊢ 𝑂 = (LPlanes‘𝐾)
13		dalem44.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
14		dalem44.z	. . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
15		dalem44.g	. . . 4 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
16	1, 9, 10, 5, 4, 11, 12, 13, 14, 15	dalem23 40899	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
17		dalem44.h	. . . 4 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
18	1, 9, 10, 5, 4, 11, 12, 13, 14, 17	dalem29 40904	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
19		dalem44.i	. . . 4 ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
20	1, 9, 10, 5, 4, 11, 12, 13, 14, 19	dalem34 40909	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴)
21	16, 18, 20	3jca 1146	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴))
22	1	dalempea 40829	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
23	1	dalemqea 40830	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
24	1	dalemrea 40831	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
25	22, 23, 24	3jca 1146	. . 3 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴))
26	25	3ad2ant1 1151	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴))
27	1, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19	dalem42 40917	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂)
28	1	dalemyeo 40835	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
29	28	3ad2ant1 1151	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ 𝑂)
30	1, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19	dalem45 40920	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ (𝐺 ∨ 𝐻))
31	1, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19	dalem46 40921	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ (𝐻 ∨ 𝐼))
32	1, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19	dalem47 40922	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ (𝐼 ∨ 𝐺))
33	30, 31, 32	3jca 1146	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)))
34	1, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19	dalem48 40923	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ (𝑃 ∨ 𝑄))
35	1, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19	dalem49 40924	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ (𝑄 ∨ 𝑅))
36	1, 9, 10, 5, 4, 11, 12, 13, 14, 15, 17, 19	dalem50 40925	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ (𝑅 ∨ 𝑃))
37	34, 35, 36	3jca 1146	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)))
38	37	3adant2 1149	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)))
39	1, 9, 10, 5, 4, 11, 12, 13, 14, 15	dalem27 40902	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (𝐺 ∨ 𝑃))
40	1, 9, 10, 5, 4, 11, 12, 13, 14, 17	dalem32 40907	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (𝐻 ∨ 𝑄))
41	1, 9, 10, 5, 4, 11, 12, 13, 14, 19	dalem36 40911	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (𝐼 ∨ 𝑅))
42	39, 40, 41	3jca 1146	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))
43		biid 264	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))) ↔ (((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))))
44		eqid 2763	. . 3 ⊢ ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((𝐺 ∨ 𝐻) ∨ 𝐼)
45		eqid 2763	. . 3 ⊢ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄))
46	43, 9, 10, 5, 11, 12, 44, 13, 45	dalemdea 40865	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
47	8, 21, 26, 27, 29, 33, 38, 42, 46	syl323anc 1427	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146   ≠ wne 2959   class class class wbr 5106  ‘cfv 6547  (class class class)co 7430  Basecbs 17402  lecple 17450  joincjn 18500  meetcmee 18501  Atomscatm 40466  HLchlt 40553  LPlanesclpl 40695

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  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 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-proset 18483  df-poset 18502  df-plt 18517  df-lub 18533  df-glb 18534  df-join 18535  df-meet 18536  df-p0 18612  df-lat 18621  df-clat 18688  df-oposet 40379  df-ol 40381  df-oml 40382  df-covers 40469  df-ats 40470  df-atl 40501  df-cvlat 40525  df-hlat 40554  df-llines 40701  df-lplanes 40702  df-lvols 40703

This theorem is used by:  dalem54  40929  dalem55  40930