Theorem dalem16 40125

Description: Lemma for dath 40182. 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 2736	. . . 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 40121	. . 3 ⊢ (𝜑 → 𝐹 ≤ (𝑌 ∧ 𝑍))
12	11	adantr 480	. 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 40119	. . . . 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 40120	. . . . 5 ⊢ (𝜑 → 𝐸 ≤ (𝑌 ∧ 𝑍))
17	1	dalemkelat 40070	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
18	1, 2, 3, 4, 5, 6, 7, 8, 13	dalemdea 40108	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
19		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
20	19, 4	atbase 39735	. . . . . . 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 40109	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
23	19, 4	atbase 39735	. . . . . . 7 ⊢ (𝐸 ∈ 𝐴 → 𝐸 ∈ (Base‘𝐾))
24	22, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (Base‘𝐾))
25	1, 6	dalemyeb 40095	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
26	1	dalemzeo 40079	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
27	19, 6	lplnbase 39980	. . . . . . . 8 ⊢ (𝑍 ∈ 𝑂 → 𝑍 ∈ (Base‘𝐾))
28	26, 27	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐾))
29	19, 5	latmcl 18406	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑍 ∈ (Base‘𝐾)) → (𝑌 ∧ 𝑍) ∈ (Base‘𝐾))
30	17, 25, 28, 29	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → (𝑌 ∧ 𝑍) ∈ (Base‘𝐾))
31	19, 2, 3	latjle12 18416	. . . . . 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 1375	. . . . 5 ⊢ (𝜑 → ((𝐷 ≤ (𝑌 ∧ 𝑍) ∧ 𝐸 ≤ (𝑌 ∧ 𝑍)) ↔ (𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍)))
33	14, 16, 32	mpbi2and 713	. . . 4 ⊢ (𝜑 → (𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍))
34	33	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍))
35	1	dalemkehl 40069	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
36	35	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐾 ∈ HL)
37	1, 2, 3, 4, 5, 6, 7, 8, 13, 15	dalemdnee 40112	. . . . . 6 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
38		eqid 2736	. . . . . . 7 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
39	3, 4, 38	llni2 39958	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝐷 ∈ 𝐴 ∧ 𝐸 ∈ 𝐴) ∧ 𝐷 ≠ 𝐸) → (𝐷 ∨ 𝐸) ∈ (LLines‘𝐾))
40	35, 18, 22, 37, 39	syl31anc 1376	. . . . 5 ⊢ (𝜑 → (𝐷 ∨ 𝐸) ∈ (LLines‘𝐾))
41	40	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝐷 ∨ 𝐸) ∈ (LLines‘𝐾))
42	1, 2, 3, 4, 5, 38, 6, 7, 8, 9	dalem15 40124	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∧ 𝑍) ∈ (LLines‘𝐾))
43	2, 38	llncmp 39968	. . . 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 1374	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → ((𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍) ↔ (𝐷 ∨ 𝐸) = (𝑌 ∧ 𝑍)))
45	34, 44	mpbid 232	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝐷 ∨ 𝐸) = (𝑌 ∧ 𝑍))
46	12, 45	breqtrrd 5113	1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐹 ≤ (𝐷 ∨ 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  lecple 17227  joincjn 18277  meetcmee 18278  Latclat 18397  Atomscatm 39709  HLchlt 39796  LLinesclln 39937  LPlanesclpl 39938

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-proset 18260  df-poset 18279  df-plt 18294  df-lub 18310  df-glb 18311  df-join 18312  df-meet 18313  df-p0 18389  df-lat 18398  df-clat 18465  df-oposet 39622  df-ol 39624  df-oml 39625  df-covers 39712  df-ats 39713  df-atl 39744  df-cvlat 39768  df-hlat 39797  df-llines 39944  df-lplanes 39945  df-lvols 39946

This theorem is referenced by:  dalem63  40181