Theorem dalem16 36296

Description: Lemma for dath 36353. 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 2793	. . . 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 36292	. . 3 ⊢ (𝜑 → 𝐹 ≤ (𝑌 ∧ 𝑍))
12	11	adantr 481	. 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 36290	. . . . 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 36291	. . . . 5 ⊢ (𝜑 → 𝐸 ≤ (𝑌 ∧ 𝑍))
17	1	dalemkelat 36241	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
18	1, 2, 3, 4, 5, 6, 7, 8, 13	dalemdea 36279	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
19		eqid 2793	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
20	19, 4	atbase 35906	. . . . . . 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 36280	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
23	19, 4	atbase 35906	. . . . . . 7 ⊢ (𝐸 ∈ 𝐴 → 𝐸 ∈ (Base‘𝐾))
24	22, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (Base‘𝐾))
25	1, 6	dalemyeb 36266	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
26	1	dalemzeo 36250	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
27	19, 6	lplnbase 36151	. . . . . . . 8 ⊢ (𝑍 ∈ 𝑂 → 𝑍 ∈ (Base‘𝐾))
28	26, 27	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐾))
29	19, 5	latmcl 17479	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑍 ∈ (Base‘𝐾)) → (𝑌 ∧ 𝑍) ∈ (Base‘𝐾))
30	17, 25, 28, 29	syl3anc 1362	. . . . . 6 ⊢ (𝜑 → (𝑌 ∧ 𝑍) ∈ (Base‘𝐾))
31	19, 2, 3	latjle12 17489	. . . . . 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 1363	. . . . 5 ⊢ (𝜑 → ((𝐷 ≤ (𝑌 ∧ 𝑍) ∧ 𝐸 ≤ (𝑌 ∧ 𝑍)) ↔ (𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍)))
33	14, 16, 32	mpbi2and 708	. . . 4 ⊢ (𝜑 → (𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍))
34	33	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍))
35	1	dalemkehl 36240	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL)
36	35	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐾 ∈ HL)
37	1, 2, 3, 4, 5, 6, 7, 8, 13, 15	dalemdnee 36283	. . . . . 6 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
38		eqid 2793	. . . . . . 7 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
39	3, 4, 38	llni2 36129	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝐷 ∈ 𝐴 ∧ 𝐸 ∈ 𝐴) ∧ 𝐷 ≠ 𝐸) → (𝐷 ∨ 𝐸) ∈ (LLines‘𝐾))
40	35, 18, 22, 37, 39	syl31anc 1364	. . . . 5 ⊢ (𝜑 → (𝐷 ∨ 𝐸) ∈ (LLines‘𝐾))
41	40	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝐷 ∨ 𝐸) ∈ (LLines‘𝐾))
42	1, 2, 3, 4, 5, 38, 6, 7, 8, 9	dalem15 36295	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∧ 𝑍) ∈ (LLines‘𝐾))
43	2, 38	llncmp 36139	. . . 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 1362	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → ((𝐷 ∨ 𝐸) ≤ (𝑌 ∧ 𝑍) ↔ (𝐷 ∨ 𝐸) = (𝑌 ∧ 𝑍)))
45	34, 44	mpbid 233	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝐷 ∨ 𝐸) = (𝑌 ∧ 𝑍))
46	12, 45	breqtrrd 4984	1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐹 ≤ (𝐷 ∨ 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1078   = wceq 1520   ∈ wcel 2079   ≠ wne 2982   class class class wbr 4956  ‘cfv 6217  (class class class)co 7007  Basecbs 16300  lecple 16389  joincjn 17371  meetcmee 17372  Latclat 17472  Atomscatm 35880  HLchlt 35967  LLinesclln 36108  LPlanesclpl 36109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-riota 6968  df-ov 7010  df-oprab 7011  df-proset 17355  df-poset 17373  df-plt 17385  df-lub 17401  df-glb 17402  df-join 17403  df-meet 17404  df-p0 17466  df-lat 17473  df-clat 17535  df-oposet 35793  df-ol 35795  df-oml 35796  df-covers 35883  df-ats 35884  df-atl 35915  df-cvlat 35939  df-hlat 35968  df-llines 36115  df-lplanes 36116  df-lvols 36117

This theorem is referenced by:  dalem63  36352