Theorem dalemcea 40152

Description: Lemma for dath 40228. 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 40117	. . 3 ⊢ (𝜑 → 𝐾 ∈ OP)
3		dalemc.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 3	dalemceb 40130	. . 3 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
5	1	dalemkehl 40115	. . . 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 40145	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾))
11	1	dalemqea 40119	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
12	1	dalemtea 40122	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
13	1, 6, 7, 3, 8, 9	dalemqnet 40144	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑇)
14		eqid 2739	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
15	7, 3, 14	llni2 40004	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑄 ≠ 𝑇) → (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾))
16	5, 11, 12, 13, 15	syl31anc 1381	. . . 4 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾))
17	1, 6, 7, 3, 8, 9	dalem1 40151	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ≠ (𝑄 ∨ 𝑇))
18	1	dalem-clpjq 40129	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
19	1, 7, 3	dalempjqeb 40137	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
20		eqid 2739	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
21		eqid 2739	. . . . . . . . . . . 12 ⊢ (0.‘𝐾) = (0.‘𝐾)
22	20, 6, 21	op0le 39678	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (0.‘𝐾) ≤ (𝑃 ∨ 𝑄))
23	2, 19, 22	syl2anc 590	. . . . . . . . . 10 ⊢ (𝜑 → (0.‘𝐾) ≤ (𝑃 ∨ 𝑄))
24		breq1 5075	. . . . . . . . . 10 ⊢ (𝐶 = (0.‘𝐾) → (𝐶 ≤ (𝑃 ∨ 𝑄) ↔ (0.‘𝐾) ≤ (𝑃 ∨ 𝑄)))
25	23, 24	syl5ibrcom 248	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 = (0.‘𝐾) → 𝐶 ≤ (𝑃 ∨ 𝑄)))
26	25	necon3bd 2948	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝐶 ≤ (𝑃 ∨ 𝑄) → 𝐶 ≠ (0.‘𝐾)))
27	18, 26	mpd 15	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ (0.‘𝐾))
28		eqid 2739	. . . . . . . . 9 ⊢ (lt‘𝐾) = (lt‘𝐾)
29	20, 28, 21	opltn0 39682	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝐶 ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)𝐶 ↔ 𝐶 ≠ (0.‘𝐾)))
30	2, 4, 29	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → ((0.‘𝐾)(lt‘𝐾)𝐶 ↔ 𝐶 ≠ (0.‘𝐾)))
31	27, 30	mpbird 258	. . . . . 6 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)𝐶)
32	1	dalemclpjs 40126	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
33	1	dalemclqjt 40127	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
34	1	dalemkelat 40116	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
35	1	dalempea 40118	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
36	1	dalemsea 40121	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
37	20, 7, 3	hlatjcl 39859	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
38	5, 35, 36, 37	syl3anc 1379	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
39	20, 7, 3	hlatjcl 39859	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
40	5, 11, 12, 39	syl3anc 1379	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
41		eqid 2739	. . . . . . . . 9 ⊢ (meet‘𝐾) = (meet‘𝐾)
42	20, 6, 41	latlem12 18423	. . . . . . . 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 1380	. . . . . . 7 ⊢ (𝜑 → ((𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇))))
44	32, 33, 43	mpbi2and 718	. . . . . 6 ⊢ (𝜑 → 𝐶 ≤ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)))
45		opposet 39673	. . . . . . . 8 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)
46	2, 45	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Poset)
47	20, 21	op0cl 39676	. . . . . . . 8 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
48	2, 47	syl 17	. . . . . . 7 ⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
49	20, 41	latmcl 18397	. . . . . . . 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 1379	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ (Base‘𝐾))
51	20, 6, 28	pltletr 18298	. . . . . . 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 1380	. . . . . 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 705	. . . . 5 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)))
54	20, 28, 21	opltn0 39682	. . . . . 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 590	. . . . 5 ⊢ (𝜑 → ((0.‘𝐾)(lt‘𝐾)((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ↔ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ≠ (0.‘𝐾)))
56	53, 55	mpbid 233	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ≠ (0.‘𝐾))
57	41, 21, 3, 14	2llnmat 40016	. . . 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 1386	. . 3 ⊢ (𝜑 → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ 𝐴)
59	20, 6, 21, 3	leat2 39786	. . 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 1386	. 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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  lecple 17218  Posetcpo 18264  ltcplt 18265  joincjn 18268  meetcmee 18269  0.cp0 18378  Latclat 18388  OPcops 39664  Atomscatm 39755  HLchlt 39842  LLinesclln 39983  LPlanesclpl 39984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-proset 18251  df-poset 18270  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-lat 18389  df-clat 18456  df-oposet 39668  df-ol 39670  df-oml 39671  df-covers 39758  df-ats 39759  df-atl 39790  df-cvlat 39814  df-hlat 39843  df-llines 39990  df-lplanes 39991

This theorem is referenced by:  dalem2  40153  dalem5  40159  dalem-cly  40163  dalem9  40164  dalem19  40174  dalem21  40186  dalem25  40190