Theorem dalemcea 37674

Description: Lemma for dath 37750. Frequently-used utility lemma. Here we show that 𝐶 must be an atom. This is an assumption in most presentations of Desargues's theorem; instead, we assume only the 𝐶 is a lattice element, in order to make later substitutions for 𝐶 easier. (Contributed by NM, 23-Sep-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)
dalem1.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem1.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)

Assertion

Ref	Expression
dalemcea	⊢ (𝜑 → 𝐶 ∈ 𝐴)

Proof of Theorem dalemcea

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	1	dalemkeop 37639	. . 3 ⊢ (𝜑 → 𝐾 ∈ OP)
3		dalemc.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 3	dalemceb 37652	. . 3 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
5	1	dalemkehl 37637	. . . 4 ⊢ (𝜑 → 𝐾 ∈ HL)
6		dalemc.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
7		dalemc.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
8		dalem1.o	. . . . 5 ⊢ 𝑂 = (LPlanes‘𝐾)
9		dalem1.y	. . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
10	1, 6, 7, 3, 8, 9	dalempjsen 37667	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾))
11	1	dalemqea 37641	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
12	1	dalemtea 37644	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
13	1, 6, 7, 3, 8, 9	dalemqnet 37666	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑇)
14		eqid 2738	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
15	7, 3, 14	llni2 37526	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑄 ≠ 𝑇) → (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾))
16	5, 11, 12, 13, 15	syl31anc 1372	. . . 4 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾))
17	1, 6, 7, 3, 8, 9	dalem1 37673	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ≠ (𝑄 ∨ 𝑇))
18	1	dalem-clpjq 37651	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
19	1, 7, 3	dalempjqeb 37659	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
20		eqid 2738	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
21		eqid 2738	. . . . . . . . . . . 12 ⊢ (0.‘𝐾) = (0.‘𝐾)
22	20, 6, 21	op0le 37200	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (0.‘𝐾) ≤ (𝑃 ∨ 𝑄))
23	2, 19, 22	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (0.‘𝐾) ≤ (𝑃 ∨ 𝑄))
24		breq1 5077	. . . . . . . . . 10 ⊢ (𝐶 = (0.‘𝐾) → (𝐶 ≤ (𝑃 ∨ 𝑄) ↔ (0.‘𝐾) ≤ (𝑃 ∨ 𝑄)))
25	23, 24	syl5ibrcom 246	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 = (0.‘𝐾) → 𝐶 ≤ (𝑃 ∨ 𝑄)))
26	25	necon3bd 2957	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝐶 ≤ (𝑃 ∨ 𝑄) → 𝐶 ≠ (0.‘𝐾)))
27	18, 26	mpd 15	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ (0.‘𝐾))
28		eqid 2738	. . . . . . . . 9 ⊢ (lt‘𝐾) = (lt‘𝐾)
29	20, 28, 21	opltn0 37204	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝐶 ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)𝐶 ↔ 𝐶 ≠ (0.‘𝐾)))
30	2, 4, 29	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((0.‘𝐾)(lt‘𝐾)𝐶 ↔ 𝐶 ≠ (0.‘𝐾)))
31	27, 30	mpbird 256	. . . . . 6 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)𝐶)
32	1	dalemclpjs 37648	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
33	1	dalemclqjt 37649	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
34	1	dalemkelat 37638	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
35	1	dalempea 37640	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
36	1	dalemsea 37643	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
37	20, 7, 3	hlatjcl 37381	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
38	5, 35, 36, 37	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
39	20, 7, 3	hlatjcl 37381	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
40	5, 11, 12, 39	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
41		eqid 2738	. . . . . . . . 9 ⊢ (meet‘𝐾) = (meet‘𝐾)
42	20, 6, 41	latlem12 18184	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))) → ((𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇))))
43	34, 4, 38, 40, 42	syl13anc 1371	. . . . . . 7 ⊢ (𝜑 → ((𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇))))
44	32, 33, 43	mpbi2and 709	. . . . . 6 ⊢ (𝜑 → 𝐶 ≤ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)))
45		opposet 37195	. . . . . . . 8 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)
46	2, 45	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Poset)
47	20, 21	op0cl 37198	. . . . . . . 8 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
48	2, 47	syl 17	. . . . . . 7 ⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
49	20, 41	latmcl 18158	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ (Base‘𝐾))
50	34, 38, 40, 49	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ (Base‘𝐾))
51	20, 6, 28	pltletr 18061	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ ((0.‘𝐾) ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ (Base‘𝐾))) → (((0.‘𝐾)(lt‘𝐾)𝐶 ∧ 𝐶 ≤ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇))) → (0.‘𝐾)(lt‘𝐾)((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇))))
52	46, 48, 4, 50, 51	syl13anc 1371	. . . . . 6 ⊢ (𝜑 → (((0.‘𝐾)(lt‘𝐾)𝐶 ∧ 𝐶 ≤ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇))) → (0.‘𝐾)(lt‘𝐾)((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇))))
53	31, 44, 52	mp2and 696	. . . . 5 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)))
54	20, 28, 21	opltn0 37204	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ↔ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ≠ (0.‘𝐾)))
55	2, 50, 54	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((0.‘𝐾)(lt‘𝐾)((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ↔ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ≠ (0.‘𝐾)))
56	53, 55	mpbid 231	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ≠ (0.‘𝐾))
57	41, 21, 3, 14	2llnmat 37538	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾)) ∧ ((𝑃 ∨ 𝑆) ≠ (𝑄 ∨ 𝑇) ∧ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ≠ (0.‘𝐾))) → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ 𝐴)
58	5, 10, 16, 17, 56, 57	syl32anc 1377	. . 3 ⊢ (𝜑 → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ 𝐴)
59	20, 6, 21, 3	leat2 37308	. . 3 ⊢ (((𝐾 ∈ OP ∧ 𝐶 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ 𝐴) ∧ (𝐶 ≠ (0.‘𝐾) ∧ 𝐶 ≤ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)))) → 𝐶 = ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)))
60	2, 4, 58, 27, 44, 59	syl32anc 1377	. 2 ⊢ (𝜑 → 𝐶 = ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)))
61	60, 58	eqeltrd 2839	1 ⊢ (𝜑 → 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  lecple 16969  Posetcpo 18025  ltcplt 18026  joincjn 18029  meetcmee 18030  0.cp0 18141  Latclat 18149  OPcops 37186  Atomscatm 37277  HLchlt 37364  LLinesclln 37505  LPlanesclpl 37506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-lat 18150  df-clat 18217  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-llines 37512  df-lplanes 37513

This theorem is referenced by:  dalem2  37675  dalem5  37681  dalem-cly  37685  dalem9  37686  dalem19  37696  dalem21  37708  dalem25  37712