Theorem dalem5 40441

Description: Lemma for dath 40510. 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 2763	. 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 40398	. 2 ⊢ (𝜑 → 𝐾 ∈ Lat)
5		dalemc.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6	3, 5	dalemueb 40418	. 2 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
7	3	dalemkehl 40397	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
8	3	dalemrea 40402	. . 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 40434	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
13	1, 9, 5	hlatjcl 40141	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝑅 ∨ 𝐶) ∈ (Base‘𝐾))
14	7, 8, 12, 13	syl3anc 1398	. 2 ⊢ (𝜑 → (𝑅 ∨ 𝐶) ∈ (Base‘𝐾))
15		dalem5.w	. . 3 ⊢ 𝑊 = (𝑌 ∨ 𝐶)
16	3, 10	dalemyeb 40423	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
17	3, 5	dalemceb 40412	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
18	1, 9	latjcl 18490	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾))
19	4, 16, 17, 18	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾))
20	15, 19	eqeltrid 2867	. 2 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
21	3	dalemclrju 40410	. . 3 ⊢ (𝜑 → 𝐶 ≤ (𝑅 ∨ 𝑈))
22	3	dalemuea 40405	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
23	3	dalempea 40400	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
24		simp313 1341	. . . . . 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 220	. . . . 5 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑅 ∨ 𝑃))
26	2, 9, 5	atnlej1 40153	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) → 𝐶 ≠ 𝑅)
27	7, 12, 8, 23, 25, 26	syl131anc 1410	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝑅)
28	2, 9, 5	hlatexch1 40169	. . . 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 1410	. . 3 ⊢ (𝜑 → (𝐶 ≤ (𝑅 ∨ 𝑈) → 𝑈 ≤ (𝑅 ∨ 𝐶)))
30	21, 29	mpd 16	. 2 ⊢ (𝜑 → 𝑈 ≤ (𝑅 ∨ 𝐶))
31	3, 9, 5	dalempjqeb 40419	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
32	3, 5	dalemreb 40415	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
33	1, 2, 9	latlej2 18500	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
34	4, 31, 32, 33	syl3anc 1398	. . . . 5 ⊢ (𝜑 → 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
35	34, 11	breqtrrdi 5153	. . . 4 ⊢ (𝜑 → 𝑅 ≤ 𝑌)
36	1, 2, 9	latjlej1 18504	. . . . 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 1399	. . . 4 ⊢ (𝜑 → (𝑅 ≤ 𝑌 → (𝑅 ∨ 𝐶) ≤ (𝑌 ∨ 𝐶)))
38	35, 37	mpd 16	. . 3 ⊢ (𝜑 → (𝑅 ∨ 𝐶) ≤ (𝑌 ∨ 𝐶))
39	38, 15	breqtrrdi 5153	. 2 ⊢ (𝜑 → (𝑅 ∨ 𝐶) ≤ 𝑊)
40	1, 2, 4, 6, 14, 20, 30, 39	lattrd 18497	1 ⊢ (𝜑 → 𝑈 ≤ 𝑊)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  lecple 17312  joincjn 18362  Latclat 18482  Atomscatm 40037  HLchlt 40124  LPlanesclpl 40266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-proset 18345  df-poset 18364  df-plt 18379  df-lub 18395  df-glb 18396  df-join 18397  df-meet 18398  df-p0 18474  df-lat 18483  df-clat 18550  df-oposet 39950  df-ol 39952  df-oml 39953  df-covers 40040  df-ats 40041  df-atl 40072  df-cvlat 40096  df-hlat 40125  df-llines 40272  df-lplanes 40273

This theorem is referenced by:  dalem6  40442  dalem8  40444