Theorem dalemcea 39699

Description: Lemma for dath 39775. 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 39664	. . 3 ⊢ (𝜑 → 𝐾 ∈ OP)
3		dalemc.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 3	dalemceb 39677	. . 3 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
5	1	dalemkehl 39662	. . . 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 39692	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾))
11	1	dalemqea 39666	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
12	1	dalemtea 39669	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
13	1, 6, 7, 3, 8, 9	dalemqnet 39691	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑇)
14		eqid 2731	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
15	7, 3, 14	llni2 39551	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑄 ≠ 𝑇) → (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾))
16	5, 11, 12, 13, 15	syl31anc 1375	. . . 4 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾))
17	1, 6, 7, 3, 8, 9	dalem1 39698	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ≠ (𝑄 ∨ 𝑇))
18	1	dalem-clpjq 39676	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
19	1, 7, 3	dalempjqeb 39684	. . . . . . . . . . 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 39225	. . . . . . . . . . 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 5089	. . . . . . . . . 10 ⊢ (𝐶 = (0.‘𝐾) → (𝐶 ≤ (𝑃 ∨ 𝑄) ↔ (0.‘𝐾) ≤ (𝑃 ∨ 𝑄)))
25	23, 24	syl5ibrcom 247	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 = (0.‘𝐾) → 𝐶 ≤ (𝑃 ∨ 𝑄)))
26	25	necon3bd 2942	. . . . . . . 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 39229	. . . . . . . 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 257	. . . . . 6 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)𝐶)
32	1	dalemclpjs 39673	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
33	1	dalemclqjt 39674	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
34	1	dalemkelat 39663	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
35	1	dalempea 39665	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
36	1	dalemsea 39668	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
37	20, 7, 3	hlatjcl 39406	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
38	5, 35, 36, 37	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
39	20, 7, 3	hlatjcl 39406	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
40	5, 11, 12, 39	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
41		eqid 2731	. . . . . . . . 9 ⊢ (meet‘𝐾) = (meet‘𝐾)
42	20, 6, 41	latlem12 18367	. . . . . . . 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 1374	. . . . . . 7 ⊢ (𝜑 → ((𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇))))
44	32, 33, 43	mpbi2and 712	. . . . . 6 ⊢ (𝜑 → 𝐶 ≤ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)))
45		opposet 39220	. . . . . . . 8 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)
46	2, 45	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Poset)
47	20, 21	op0cl 39223	. . . . . . . 8 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
48	2, 47	syl 17	. . . . . . 7 ⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
49	20, 41	latmcl 18341	. . . . . . . 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 1373	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ (Base‘𝐾))
51	20, 6, 28	pltletr 18242	. . . . . . 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 1374	. . . . . 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 699	. . . . 5 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)))
54	20, 28, 21	opltn0 39229	. . . . . 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 232	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ≠ (0.‘𝐾))
57	41, 21, 3, 14	2llnmat 39563	. . . 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 1380	. . 3 ⊢ (𝜑 → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ 𝐴)
59	20, 6, 21, 3	leat2 39333	. . 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 1380	. 2 ⊢ (𝜑 → 𝐶 = ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)))
61	60, 58	eqeltrd 2831	1 ⊢ (𝜑 → 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928   class class class wbr 5086  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  lecple 17163  Posetcpo 18208  ltcplt 18209  joincjn 18212  meetcmee 18213  0.cp0 18322  Latclat 18332  OPcops 39211  Atomscatm 39302  HLchlt 39389  LLinesclln 39530  LPlanesclpl 39531

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-proset 18195  df-poset 18214  df-plt 18229  df-lub 18245  df-glb 18246  df-join 18247  df-meet 18248  df-p0 18324  df-lat 18333  df-clat 18400  df-oposet 39215  df-ol 39217  df-oml 39218  df-covers 39305  df-ats 39306  df-atl 39337  df-cvlat 39361  df-hlat 39390  df-llines 39537  df-lplanes 39538

This theorem is referenced by:  dalem2  39700  dalem5  39706  dalem-cly  39710  dalem9  39711  dalem19  39721  dalem21  39733  dalem25  39737