Theorem dalem5 35466

Description: Lemma for dath 35535. Atom 𝑈 (in plane 𝑍 = 𝑆𝑇𝑈) belongs to the 3-dimensional volume formed by 𝑌 and 𝐶. (Contributed by NM, 21-Jul-2012.)

Hypotheses

Ref	Expression
dalema.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)))))
dalemc.l	⊢ ≤ = (le‘𝐾)
dalemc.j	⊢ ∨ = (join‘𝐾)
dalemc.a	⊢ 𝐴 = (Atoms‘𝐾)
dalem5.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem5.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem5.w	⊢ 𝑊 = (𝑌 ∨ 𝐶)

Assertion

Ref	Expression
dalem5	⊢ (𝜑 → 𝑈 ≤ 𝑊)

Proof of Theorem dalem5

Step	Hyp	Ref	Expression
1		eqid 2817	. 2 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		dalemc.l	. 2 ⊢ ≤ = (le‘𝐾)
3		dalema.ph	. . 3 ⊢ (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	dalemkelat 35423	. 2 ⊢ (𝜑 → 𝐾 ∈ Lat)
5		dalemc.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6	3, 5	dalemueb 35443	. 2 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
7	3	dalemkehl 35422	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
8	3	dalemrea 35427	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
9		dalemc.j	. . . 4 ⊢ ∨ = (join‘𝐾)
10		dalem5.o	. . . 4 ⊢ 𝑂 = (LPlanes‘𝐾)
11		dalem5.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
12	3, 2, 9, 5, 10, 11	dalemcea 35459	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
13	1, 9, 5	hlatjcl 35166	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝑅 ∨ 𝐶) ∈ (Base‘𝐾))
14	7, 8, 12, 13	syl3anc 1483	. 2 ⊢ (𝜑 → (𝑅 ∨ 𝐶) ∈ (Base‘𝐾))
15		dalem5.w	. . 3 ⊢ 𝑊 = (𝑌 ∨ 𝐶)
16	3, 10	dalemyeb 35448	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
17	3, 5	dalemceb 35437	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
18	1, 9	latjcl 17276	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾))
19	4, 16, 17, 18	syl3anc 1483	. . 3 ⊢ (𝜑 → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾))
20	15, 19	syl5eqel 2900	. 2 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
21	3	dalemclrju 35435	. . 3 ⊢ (𝜑 → 𝐶 ≤ (𝑅 ∨ 𝑈))
22	3	dalemuea 35430	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
23	3	dalempea 35425	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
24		simp313 1414	. . . . . 6 ⊢ ((((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)))) → ¬ 𝐶 ≤ (𝑅 ∨ 𝑃))
25	3, 24	sylbi 208	. . . . 5 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑅 ∨ 𝑃))
26	2, 9, 5	atnlej1 35178	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) → 𝐶 ≠ 𝑅)
27	7, 12, 8, 23, 25, 26	syl131anc 1495	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝑅)
28	2, 9, 5	hlatexch1 35194	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝐶 ≠ 𝑅) → (𝐶 ≤ (𝑅 ∨ 𝑈) → 𝑈 ≤ (𝑅 ∨ 𝐶)))
29	7, 12, 22, 8, 27, 28	syl131anc 1495	. . 3 ⊢ (𝜑 → (𝐶 ≤ (𝑅 ∨ 𝑈) → 𝑈 ≤ (𝑅 ∨ 𝐶)))
30	21, 29	mpd 15	. 2 ⊢ (𝜑 → 𝑈 ≤ (𝑅 ∨ 𝐶))
31	3, 9, 5	dalempjqeb 35444	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
32	3, 5	dalemreb 35440	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
33	1, 2, 9	latlej2 17286	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
34	4, 31, 32, 33	syl3anc 1483	. . . . 5 ⊢ (𝜑 → 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
35	34, 11	syl6breqr 4897	. . . 4 ⊢ (𝜑 → 𝑅 ≤ 𝑌)
36	1, 2, 9	latjlej1 17290	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾))) → (𝑅 ≤ 𝑌 → (𝑅 ∨ 𝐶) ≤ (𝑌 ∨ 𝐶)))
37	4, 32, 16, 17, 36	syl13anc 1484	. . . 4 ⊢ (𝜑 → (𝑅 ≤ 𝑌 → (𝑅 ∨ 𝐶) ≤ (𝑌 ∨ 𝐶)))
38	35, 37	mpd 15	. . 3 ⊢ (𝜑 → (𝑅 ∨ 𝐶) ≤ (𝑌 ∨ 𝐶))
39	38, 15	syl6breqr 4897	. 2 ⊢ (𝜑 → (𝑅 ∨ 𝐶) ≤ 𝑊)
40	1, 2, 4, 6, 14, 20, 30, 39	lattrd 17283	1 ⊢ (𝜑 → 𝑈 ≤ 𝑊)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1100   = wceq 1637   ∈ wcel 2157   ≠ wne 2989   class class class wbr 4855  ‘cfv 6111  (class class class)co 6884  Basecbs 16088  lecple 16180  joincjn 17169  Latclat 17270  Atomscatm 35062  HLchlt 35149  LPlanesclpl 35291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-riota 6845  df-ov 6887  df-oprab 6888  df-proset 17153  df-poset 17171  df-plt 17183  df-lub 17199  df-glb 17200  df-join 17201  df-meet 17202  df-p0 17264  df-lat 17271  df-clat 17333  df-oposet 34975  df-ol 34977  df-oml 34978  df-covers 35065  df-ats 35066  df-atl 35097  df-cvlat 35121  df-hlat 35150  df-llines 35297  df-lplanes 35298

This theorem is referenced by:  dalem6  35467  dalem8  35469