Theorem dalem24 39680

Description: Lemma for dath 39719. Show that auxiliary atom 𝐺 is outside of plane 𝑌. (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))))
dalem23.m	⊢ ∧ = (meet‘𝐾)
dalem23.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem23.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem23.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem23.g	⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))

Assertion

Ref	Expression
dalem24	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐺 ≤ 𝑌)

Proof of Theorem dalem24

Step	Hyp	Ref	Expression
1		dalem23.g	. . . . 5 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
2	1	oveq1i 7359	. . . 4 ⊢ (𝐺 ∧ 𝑌) = (((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∧ 𝑌)
3		dalem.ph	. . . . . . . 8 ⊢ (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)))))
4	3	dalemkehl 39606	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ HL)
5		hlol 39344	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ OL)
7	6	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ OL)
8	4	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
9		dalem.ps	. . . . . . . 8 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
10	9	dalemccea 39666	. . . . . . 7 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
11	10	3ad2ant3 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
12	3	dalempea 39609	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
13	12	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ 𝐴)
14		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
15		dalem.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
16		dalem.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
17	14, 15, 16	hlatjcl 39350	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
18	8, 11, 13, 17	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
19	9	dalemddea 39667	. . . . . . 7 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
20	19	3ad2ant3 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ 𝐴)
21	3	dalemsea 39612	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
22	21	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ∈ 𝐴)
23	14, 15, 16	hlatjcl 39350	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾))
24	8, 20, 22, 23	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾))
25		dalem23.o	. . . . . . 7 ⊢ 𝑂 = (LPlanes‘𝐾)
26	3, 25	dalemyeb 39632	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
27	26	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
28		dalem23.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
29	14, 28	latmmdir 39218	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ ((𝑐 ∨ 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → (((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∧ 𝑌) = (((𝑐 ∨ 𝑃) ∧ 𝑌) ∧ ((𝑑 ∨ 𝑆) ∧ 𝑌)))
30	7, 18, 24, 27, 29	syl13anc 1374	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∧ 𝑌) = (((𝑐 ∨ 𝑃) ∧ 𝑌) ∧ ((𝑑 ∨ 𝑆) ∧ 𝑌)))
31	2, 30	eqtrid 2776	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∧ 𝑌) = (((𝑐 ∨ 𝑃) ∧ 𝑌) ∧ ((𝑑 ∨ 𝑆) ∧ 𝑌)))
32	15, 16	hlatjcom 39351	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑐 ∨ 𝑃) = (𝑃 ∨ 𝑐))
33	8, 11, 13, 32	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑃) = (𝑃 ∨ 𝑐))
34	33	oveq1d 7364	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑃) ∧ 𝑌) = ((𝑃 ∨ 𝑐) ∧ 𝑌))
35		dalem.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
36		dalem23.y	. . . . . . . 8 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
37	3, 35, 15, 16, 25, 36	dalemply 39637	. . . . . . 7 ⊢ (𝜑 → 𝑃 ≤ 𝑌)
38	37	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ 𝑌)
39	9	dalem-ccly 39668	. . . . . . 7 ⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌)
40	39	3ad2ant3 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ 𝑌)
41	14, 35, 15, 28, 16	2atjm 39428	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ∧ 𝑌 ∈ (Base‘𝐾)) ∧ (𝑃 ≤ 𝑌 ∧ ¬ 𝑐 ≤ 𝑌)) → ((𝑃 ∨ 𝑐) ∧ 𝑌) = 𝑃)
42	8, 13, 11, 27, 38, 40, 41	syl132anc 1390	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑐) ∧ 𝑌) = 𝑃)
43	34, 42	eqtrd 2764	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑃) ∧ 𝑌) = 𝑃)
44	15, 16	hlatjcom 39351	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) = (𝑆 ∨ 𝑑))
45	8, 20, 22, 44	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑆) = (𝑆 ∨ 𝑑))
46	45	oveq1d 7364	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑑 ∨ 𝑆) ∧ 𝑌) = ((𝑆 ∨ 𝑑) ∧ 𝑌))
47		dalem23.z	. . . . . . . 8 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
48	3, 35, 15, 16, 47	dalemsly 39638	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌)
49	48	3adant3 1132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ≤ 𝑌)
50	9	dalem-ddly 39669	. . . . . . 7 ⊢ (𝜓 → ¬ 𝑑 ≤ 𝑌)
51	50	3ad2ant3 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑑 ≤ 𝑌)
52	14, 35, 15, 28, 16	2atjm 39428	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ∧ 𝑌 ∈ (Base‘𝐾)) ∧ (𝑆 ≤ 𝑌 ∧ ¬ 𝑑 ≤ 𝑌)) → ((𝑆 ∨ 𝑑) ∧ 𝑌) = 𝑆)
53	8, 22, 20, 27, 49, 51, 52	syl132anc 1390	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑆 ∨ 𝑑) ∧ 𝑌) = 𝑆)
54	46, 53	eqtrd 2764	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑑 ∨ 𝑆) ∧ 𝑌) = 𝑆)
55	43, 54	oveq12d 7367	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝑐 ∨ 𝑃) ∧ 𝑌) ∧ ((𝑑 ∨ 𝑆) ∧ 𝑌)) = (𝑃 ∧ 𝑆))
56	3, 35, 15, 16, 25, 36	dalempnes 39634	. . . . 5 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
57		hlatl 39343	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
58	4, 57	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ AtLat)
59		eqid 2729	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
60	28, 59, 16	atnem0 39301	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ≠ 𝑆 ↔ (𝑃 ∧ 𝑆) = (0.‘𝐾)))
61	58, 12, 21, 60	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑃 ≠ 𝑆 ↔ (𝑃 ∧ 𝑆) = (0.‘𝐾)))
62	56, 61	mpbid 232	. . . 4 ⊢ (𝜑 → (𝑃 ∧ 𝑆) = (0.‘𝐾))
63	62	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∧ 𝑆) = (0.‘𝐾))
64	31, 55, 63	3eqtrd 2768	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∧ 𝑌) = (0.‘𝐾))
65	58	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ AtLat)
66	3, 35, 15, 16, 9, 28, 25, 36, 47, 1	dalem23 39679	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
67	14, 35, 28, 59, 16	atnle 39300	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝐺 ∈ 𝐴 ∧ 𝑌 ∈ (Base‘𝐾)) → (¬ 𝐺 ≤ 𝑌 ↔ (𝐺 ∧ 𝑌) = (0.‘𝐾)))
68	65, 66, 27, 67	syl3anc 1373	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (¬ 𝐺 ≤ 𝑌 ↔ (𝐺 ∧ 𝑌) = (0.‘𝐾)))
69	64, 68	mpbird 257	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐺 ≤ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  lecple 17168  joincjn 18217  meetcmee 18218  0.cp0 18327  OLcol 39157  Atomscatm 39246  AtLatcal 39247  HLchlt 39333  LPlanesclpl 39475

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-proset 18200  df-poset 18219  df-plt 18234  df-lub 18250  df-glb 18251  df-join 18252  df-meet 18253  df-p0 18329  df-lat 18338  df-clat 18405  df-oposet 39159  df-ol 39161  df-oml 39162  df-covers 39249  df-ats 39250  df-atl 39281  df-cvlat 39305  df-hlat 39334  df-llines 39481  df-lplanes 39482

This theorem is referenced by:  dalem27  39682  dalem30  39685  dalem54  39709