Theorem dalem16 37255

Description: Lemma for dath 37312. The atoms 𝐷, 𝐸, and 𝐹 form a line of perspectivity. This is Desargues's theorem for the special case where planes 𝑌 and 𝑍 are different. (Contributed by NM, 7-Aug-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)
dalem16.m	⊢ ∧ = (meet‘𝐾)
dalem16.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem16.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem16.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem16.d	⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))
dalem16.e	⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈))
dalem16.f	⊢ 𝐹 = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆))

Assertion

Ref	Expression
dalem16	⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐹 ≤ (𝐷 ∨ 𝐸))

Proof of Theorem dalem16

Step	Hyp	Ref	Expression
1		dalema.ph	. . . 4 ⊢ (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		dalemc.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		dalemc.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		dalemc.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5		dalem16.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
6		dalem16.o	. . . 4 ⊢ 𝑂 = (LPlanes‘𝐾)
7		dalem16.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
8		dalem16.z	. . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
9		eqid 2758	. . . 4 ⊢ (𝑌 ∧ 𝑍) = (𝑌 ∧ 𝑍)
10		dalem16.f	. . . 4 ⊢ 𝐹 = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	dalem12 37251	. . 3 ⊢ (𝜑 → 𝐹 ≤ (𝑌 ∧ 𝑍))
12	11	adantr 484	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐹 ≤ (𝑌 ∧ 𝑍))
13		dalem16.d	. . . . . 6 ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 13	dalem10 37249	. . . . 5 ⊢ (𝜑 → 𝐷 ≤ (𝑌 ∧ 𝑍))
15		dalem16.e	. . . . . 6 ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈))
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 15	dalem11 37250	. . . . 5 ⊢ (𝜑 → 𝐸 ≤ (𝑌 ∧ 𝑍))
17	1	dalemkelat 37200	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
18	1, 2, 3, 4, 5, 6, 7, 8, 13	dalemdea 37238	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
19		eqid 2758	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
20	19, 4	atbase 36865	. . . . . . 7 ⊢ (𝐷 ∈ 𝐴 → 𝐷 ∈ (Base‘𝐾))
21	18, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (Base‘𝐾))
22	1, 2, 3, 4, 5, 6, 7, 8, 15	dalemeea 37239	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
23	19, 4	atbase 36865	. . . . . . 7 ⊢ (𝐸 ∈ 𝐴 → 𝐸 ∈ (Base‘𝐾))
24	22, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (Base‘𝐾))
25	1, 6	dalemyeb 37225	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
26	1	dalemzeo 37209	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
27	19, 6	lplnbase 37110	. . . . . . . 8 ⊢ (𝑍 ∈ 𝑂 → 𝑍 ∈ (Base‘𝐾))
28	26, 27	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐾))
29	19, 5	latmcl 17728	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑍 ∈ (Base‘𝐾)) → (𝑌 ∧ 𝑍) ∈ (Base‘𝐾))
30	17, 25, 28, 29	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝑌 ∧ 𝑍) ∈ (Base‘𝐾))
31	19, 2, 3	latjle12 17738	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐷 ∈ (Base‘𝐾) ∧ 𝐸 ∈ (Base‘𝐾) ∧ (𝑌 ∧ 𝑍) ∈ (Base‘𝐾))) → ((𝐷 ≤ (𝑌 ∧ 𝑍) ∧ 𝐸 ≤ (𝑌 ∧ 𝑍)) ↔ (𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍)))
32	17, 21, 24, 30, 31	syl13anc 1369	. . . . 5 ⊢ (𝜑 → ((𝐷 ≤ (𝑌 ∧ 𝑍) ∧ 𝐸 ≤ (𝑌 ∧ 𝑍)) ↔ (𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍)))
33	14, 16, 32	mpbi2and 711	. . . 4 ⊢ (𝜑 → (𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍))
34	33	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍))
35	1	dalemkehl 37199	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
36	35	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐾 ∈ HL)
37	1, 2, 3, 4, 5, 6, 7, 8, 13, 15	dalemdnee 37242	. . . . . 6 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
38		eqid 2758	. . . . . . 7 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
39	3, 4, 38	llni2 37088	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝐷 ∈ 𝐴 ∧ 𝐸 ∈ 𝐴) ∧ 𝐷 ≠ 𝐸) → (𝐷 ∨ 𝐸) ∈ (LLines‘𝐾))
40	35, 18, 22, 37, 39	syl31anc 1370	. . . . 5 ⊢ (𝜑 → (𝐷 ∨ 𝐸) ∈ (LLines‘𝐾))
41	40	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝐷 ∨ 𝐸) ∈ (LLines‘𝐾))
42	1, 2, 3, 4, 5, 38, 6, 7, 8, 9	dalem15 37254	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∧ 𝑍) ∈ (LLines‘𝐾))
43	2, 38	llncmp 37098	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝐷 ∨ 𝐸) ∈ (LLines‘𝐾) ∧ (𝑌 ∧ 𝑍) ∈ (LLines‘𝐾)) → ((𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍) ↔ (𝐷 ∨ 𝐸) = (𝑌 ∧ 𝑍)))
44	36, 41, 42, 43	syl3anc 1368	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → ((𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍) ↔ (𝐷 ∨ 𝐸) = (𝑌 ∧ 𝑍)))
45	34, 44	mpbid 235	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝐷 ∨ 𝐸) = (𝑌 ∧ 𝑍))
46	12, 45	breqtrrd 5060	1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐹 ≤ (𝐷 ∨ 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   class class class wbr 5032  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  lecple 16630  joincjn 17620  meetcmee 17621  Latclat 17721  Atomscatm 36839  HLchlt 36926  LLinesclln 37067  LPlanesclpl 37068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17604  df-poset 17622  df-plt 17634  df-lub 17650  df-glb 17651  df-join 17652  df-meet 17653  df-p0 17715  df-lat 17722  df-clat 17784  df-oposet 36752  df-ol 36754  df-oml 36755  df-covers 36842  df-ats 36843  df-atl 36874  df-cvlat 36898  df-hlat 36927  df-llines 37074  df-lplanes 37075  df-lvols 37076

This theorem is referenced by:  dalem63  37311