Theorem dalemcea 38835

Description: Lemma for dath 38911. 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 38800	. . 3 ⊢ (𝜑 → 𝐾 ∈ OP)
3		dalemc.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 3	dalemceb 38813	. . 3 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
5	1	dalemkehl 38798	. . . 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 38828	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾))
11	1	dalemqea 38802	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
12	1	dalemtea 38805	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
13	1, 6, 7, 3, 8, 9	dalemqnet 38827	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑇)
14		eqid 2731	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
15	7, 3, 14	llni2 38687	. . . . 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 38834	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ≠ (𝑄 ∨ 𝑇))
18	1	dalem-clpjq 38812	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
19	1, 7, 3	dalempjqeb 38820	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
20		eqid 2731	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
21		eqid 2731	. . . . . . . . . . . 12 ⊢ (0.‘𝐾) = (0.‘𝐾)
22	20, 6, 21	op0le 38360	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (0.‘𝐾) ≤ (𝑃 ∨ 𝑄))
23	2, 19, 22	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (0.‘𝐾) ≤ (𝑃 ∨ 𝑄))
24		breq1 5151	. . . . . . . . . 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 2953	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝐶 ≤ (𝑃 ∨ 𝑄) → 𝐶 ≠ (0.‘𝐾)))
27	18, 26	mpd 15	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ (0.‘𝐾))
28		eqid 2731	. . . . . . . . 9 ⊢ (lt‘𝐾) = (lt‘𝐾)
29	20, 28, 21	opltn0 38364	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝐶 ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)𝐶 ↔ 𝐶 ≠ (0.‘𝐾)))
30	2, 4, 29	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((0.‘𝐾)(lt‘𝐾)𝐶 ↔ 𝐶 ≠ (0.‘𝐾)))
31	27, 30	mpbird 257	. . . . . 6 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)𝐶)
32	1	dalemclpjs 38809	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
33	1	dalemclqjt 38810	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
34	1	dalemkelat 38799	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
35	1	dalempea 38801	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
36	1	dalemsea 38804	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
37	20, 7, 3	hlatjcl 38541	. . . . . . . . 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 38541	. . . . . . . . 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 2731	. . . . . . . . 9 ⊢ (meet‘𝐾) = (meet‘𝐾)
42	20, 6, 41	latlem12 18424	. . . . . . . 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 38355	. . . . . . . 8 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)
46	2, 45	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Poset)
47	20, 21	op0cl 38358	. . . . . . . 8 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
48	2, 47	syl 17	. . . . . . 7 ⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
49	20, 41	latmcl 18398	. . . . . . . 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 18301	. . . . . . 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 38364	. . . . . 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 583	. . . . 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 38699	. . . 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 38468	. . 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 2832	1 ⊢ (𝜑 → 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2939   class class class wbr 5148  ‘cfv 6543  (class class class)co 7412  Basecbs 17149  lecple 17209  Posetcpo 18265  ltcplt 18266  joincjn 18269  meetcmee 18270  0.cp0 18381  Latclat 18389  OPcops 38346  Atomscatm 38437  HLchlt 38524  LLinesclln 38666  LPlanesclpl 38667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-llines 38673  df-lplanes 38674

This theorem is referenced by:  dalem2  38836  dalem5  38842  dalem-cly  38846  dalem9  38847  dalem19  38857  dalem21  38869  dalem25  38873