Theorem dalemcea 38123

Description: Lemma for dath 38199. 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 38088	. . 3 ⊢ (𝜑 → 𝐾 ∈ OP)
3		dalemc.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 3	dalemceb 38101	. . 3 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
5	1	dalemkehl 38086	. . . 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 38116	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (LLines‘𝐾))
11	1	dalemqea 38090	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
12	1	dalemtea 38093	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
13	1, 6, 7, 3, 8, 9	dalemqnet 38115	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑇)
14		eqid 2736	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
15	7, 3, 14	llni2 37975	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑄 ≠ 𝑇) → (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾))
16	5, 11, 12, 13, 15	syl31anc 1373	. . . 4 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (LLines‘𝐾))
17	1, 6, 7, 3, 8, 9	dalem1 38122	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ≠ (𝑄 ∨ 𝑇))
18	1	dalem-clpjq 38100	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
19	1, 7, 3	dalempjqeb 38108	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
20		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
21		eqid 2736	. . . . . . . . . . . 12 ⊢ (0.‘𝐾) = (0.‘𝐾)
22	20, 6, 21	op0le 37648	. . . . . . . . . . 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 5108	. . . . . . . . . 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 2736	. . . . . . . . 9 ⊢ (lt‘𝐾) = (lt‘𝐾)
29	20, 28, 21	opltn0 37652	. . . . . . . 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 38097	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
33	1	dalemclqjt 38098	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
34	1	dalemkelat 38087	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
35	1	dalempea 38089	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
36	1	dalemsea 38092	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
37	20, 7, 3	hlatjcl 37829	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
38	5, 35, 36, 37	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
39	20, 7, 3	hlatjcl 37829	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
40	5, 11, 12, 39	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
41		eqid 2736	. . . . . . . . 9 ⊢ (meet‘𝐾) = (meet‘𝐾)
42	20, 6, 41	latlem12 18355	. . . . . . . 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 1372	. . . . . . 7 ⊢ (𝜑 → ((𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇)) ↔ 𝐶 ≤ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇))))
44	32, 33, 43	mpbi2and 710	. . . . . 6 ⊢ (𝜑 → 𝐶 ≤ ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)))
45		opposet 37643	. . . . . . . 8 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)
46	2, 45	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Poset)
47	20, 21	op0cl 37646	. . . . . . . 8 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
48	2, 47	syl 17	. . . . . . 7 ⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
49	20, 41	latmcl 18329	. . . . . . . 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 1371	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ (Base‘𝐾))
51	20, 6, 28	pltletr 18232	. . . . . . 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 1372	. . . . . 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 697	. . . . 5 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)))
54	20, 28, 21	opltn0 37652	. . . . . 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 37987	. . . 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 1378	. . 3 ⊢ (𝜑 → ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)) ∈ 𝐴)
59	20, 6, 21, 3	leat2 37756	. . 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 1378	. 2 ⊢ (𝜑 → 𝐶 = ((𝑃 ∨ 𝑆)(meet‘𝐾)(𝑄 ∨ 𝑇)))
61	60, 58	eqeltrd 2838	1 ⊢ (𝜑 → 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  Posetcpo 18196  ltcplt 18197  joincjn 18200  meetcmee 18201  0.cp0 18312  Latclat 18320  OPcops 37634  Atomscatm 37725  HLchlt 37812  LLinesclln 37954  LPlanesclpl 37955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962

This theorem is referenced by:  dalem2  38124  dalem5  38130  dalem-cly  38134  dalem9  38135  dalem19  38145  dalem21  38157  dalem25  38161