Theorem dalem5 39650

Description: Lemma for dath 39719. 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 2735	. 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 39607	. 2 ⊢ (𝜑 → 𝐾 ∈ Lat)
5		dalemc.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6	3, 5	dalemueb 39627	. 2 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
7	3	dalemkehl 39606	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
8	3	dalemrea 39611	. . 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 39643	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
13	1, 9, 5	hlatjcl 39349	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝑅 ∨ 𝐶) ∈ (Base‘𝐾))
14	7, 8, 12, 13	syl3anc 1370	. 2 ⊢ (𝜑 → (𝑅 ∨ 𝐶) ∈ (Base‘𝐾))
15		dalem5.w	. . 3 ⊢ 𝑊 = (𝑌 ∨ 𝐶)
16	3, 10	dalemyeb 39632	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
17	3, 5	dalemceb 39621	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
18	1, 9	latjcl 18497	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾))
19	4, 16, 17, 18	syl3anc 1370	. . 3 ⊢ (𝜑 → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾))
20	15, 19	eqeltrid 2843	. 2 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
21	3	dalemclrju 39619	. . 3 ⊢ (𝜑 → 𝐶 ≤ (𝑅 ∨ 𝑈))
22	3	dalemuea 39614	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
23	3	dalempea 39609	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
24		simp313 1321	. . . . . 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 217	. . . . 5 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑅 ∨ 𝑃))
26	2, 9, 5	atnlej1 39362	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) → 𝐶 ≠ 𝑅)
27	7, 12, 8, 23, 25, 26	syl131anc 1382	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝑅)
28	2, 9, 5	hlatexch1 39378	. . . 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 1382	. . 3 ⊢ (𝜑 → (𝐶 ≤ (𝑅 ∨ 𝑈) → 𝑈 ≤ (𝑅 ∨ 𝐶)))
30	21, 29	mpd 15	. 2 ⊢ (𝜑 → 𝑈 ≤ (𝑅 ∨ 𝐶))
31	3, 9, 5	dalempjqeb 39628	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
32	3, 5	dalemreb 39624	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
33	1, 2, 9	latlej2 18507	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
34	4, 31, 32, 33	syl3anc 1370	. . . . 5 ⊢ (𝜑 → 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
35	34, 11	breqtrrdi 5190	. . . 4 ⊢ (𝜑 → 𝑅 ≤ 𝑌)
36	1, 2, 9	latjlej1 18511	. . . . 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 1371	. . . 4 ⊢ (𝜑 → (𝑅 ≤ 𝑌 → (𝑅 ∨ 𝐶) ≤ (𝑌 ∨ 𝐶)))
38	35, 37	mpd 15	. . 3 ⊢ (𝜑 → (𝑅 ∨ 𝐶) ≤ (𝑌 ∨ 𝐶))
39	38, 15	breqtrrdi 5190	. 2 ⊢ (𝜑 → (𝑅 ∨ 𝐶) ≤ 𝑊)
40	1, 2, 4, 6, 14, 20, 30, 39	lattrd 18504	1 ⊢ (𝜑 → 𝑈 ≤ 𝑊)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  Latclat 18489  Atomscatm 39245  HLchlt 39332  LPlanesclpl 39475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482

This theorem is referenced by:  dalem6  39651  dalem8  39653