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Theorem dalemcea 35264
Description: Lemma for dath 35340. Frequently-used utility lemma. Here we show that 𝐶 must be an atom. This is an assumption in most presentations of Desargue'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)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 (𝜑𝐶𝐴)

Proof of Theorem dalemcea
StepHypRef Expression
1 dalema.ph . . . 4 (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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)))))
21dalemkeop 35229 . . 3 (𝜑𝐾 ∈ OP)
3 dalemc.a . . . 4 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 35242 . . 3 (𝜑𝐶 ∈ (Base‘𝐾))
51dalemkehl 35227 . . . 4 (𝜑𝐾 ∈ 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)
101, 6, 7, 3, 8, 9dalempjsen 35257 . . . 4 (𝜑 → (𝑃 𝑆) ∈ (LLines‘𝐾))
111dalemqea 35231 . . . . 5 (𝜑𝑄𝐴)
121dalemtea 35234 . . . . 5 (𝜑𝑇𝐴)
131, 6, 7, 3, 8, 9dalemqnet 35256 . . . . 5 (𝜑𝑄𝑇)
14 eqid 2651 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
157, 3, 14llni2 35116 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) ∧ 𝑄𝑇) → (𝑄 𝑇) ∈ (LLines‘𝐾))
165, 11, 12, 13, 15syl31anc 1369 . . . 4 (𝜑 → (𝑄 𝑇) ∈ (LLines‘𝐾))
171, 6, 7, 3, 8, 9dalem1 35263 . . . 4 (𝜑 → (𝑃 𝑆) ≠ (𝑄 𝑇))
181dalem-clpjq 35241 . . . . . . . 8 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
191, 7, 3dalempjqeb 35249 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
20 eqid 2651 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
21 eqid 2651 . . . . . . . . . . . 12 (0.‘𝐾) = (0.‘𝐾)
2220, 6, 21op0le 34791 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (0.‘𝐾) (𝑃 𝑄))
232, 19, 22syl2anc 694 . . . . . . . . . 10 (𝜑 → (0.‘𝐾) (𝑃 𝑄))
24 breq1 4688 . . . . . . . . . 10 (𝐶 = (0.‘𝐾) → (𝐶 (𝑃 𝑄) ↔ (0.‘𝐾) (𝑃 𝑄)))
2523, 24syl5ibrcom 237 . . . . . . . . 9 (𝜑 → (𝐶 = (0.‘𝐾) → 𝐶 (𝑃 𝑄)))
2625necon3bd 2837 . . . . . . . 8 (𝜑 → (¬ 𝐶 (𝑃 𝑄) → 𝐶 ≠ (0.‘𝐾)))
2718, 26mpd 15 . . . . . . 7 (𝜑𝐶 ≠ (0.‘𝐾))
28 eqid 2651 . . . . . . . . 9 (lt‘𝐾) = (lt‘𝐾)
2920, 28, 21opltn0 34795 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝐶 ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)𝐶𝐶 ≠ (0.‘𝐾)))
302, 4, 29syl2anc 694 . . . . . . 7 (𝜑 → ((0.‘𝐾)(lt‘𝐾)𝐶𝐶 ≠ (0.‘𝐾)))
3127, 30mpbird 247 . . . . . 6 (𝜑 → (0.‘𝐾)(lt‘𝐾)𝐶)
321dalemclpjs 35238 . . . . . . 7 (𝜑𝐶 (𝑃 𝑆))
331dalemclqjt 35239 . . . . . . 7 (𝜑𝐶 (𝑄 𝑇))
341dalemkelat 35228 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
351dalempea 35230 . . . . . . . . 9 (𝜑𝑃𝐴)
361dalemsea 35233 . . . . . . . . 9 (𝜑𝑆𝐴)
3720, 7, 3hlatjcl 34971 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
385, 35, 36, 37syl3anc 1366 . . . . . . . 8 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
3920, 7, 3hlatjcl 34971 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) ∈ (Base‘𝐾))
405, 11, 12, 39syl3anc 1366 . . . . . . . 8 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
41 eqid 2651 . . . . . . . . 9 (meet‘𝐾) = (meet‘𝐾)
4220, 6, 41latlem12 17125 . . . . . . . 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))))
4334, 4, 38, 40, 42syl13anc 1368 . . . . . . 7 (𝜑 → ((𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇)) ↔ 𝐶 ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇))))
4432, 33, 43mpbi2and 976 . . . . . 6 (𝜑𝐶 ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)))
45 opposet 34786 . . . . . . . 8 (𝐾 ∈ OP → 𝐾 ∈ Poset)
462, 45syl 17 . . . . . . 7 (𝜑𝐾 ∈ Poset)
4720, 21op0cl 34789 . . . . . . . 8 (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
482, 47syl 17 . . . . . . 7 (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
4920, 41latmcl 17099 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ (Base‘𝐾))
5034, 38, 40, 49syl3anc 1366 . . . . . . 7 (𝜑 → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ (Base‘𝐾))
5120, 6, 28pltletr 17018 . . . . . . 7 ((𝐾 ∈ Poset ∧ ((0.‘𝐾) ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ (Base‘𝐾))) → (((0.‘𝐾)(lt‘𝐾)𝐶𝐶 ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇))) → (0.‘𝐾)(lt‘𝐾)((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇))))
5246, 48, 4, 50, 51syl13anc 1368 . . . . . 6 (𝜑 → (((0.‘𝐾)(lt‘𝐾)𝐶𝐶 ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇))) → (0.‘𝐾)(lt‘𝐾)((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇))))
5331, 44, 52mp2and 715 . . . . 5 (𝜑 → (0.‘𝐾)(lt‘𝐾)((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)))
5420, 28, 21opltn0 34795 . . . . . 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)))
552, 50, 54syl2anc 694 . . . . 5 (𝜑 → ((0.‘𝐾)(lt‘𝐾)((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ↔ ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ≠ (0.‘𝐾)))
5653, 55mpbid 222 . . . 4 (𝜑 → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ≠ (0.‘𝐾))
5741, 21, 3, 142llnmat 35128 . . . 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)
585, 10, 16, 17, 56, 57syl32anc 1374 . . 3 (𝜑 → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ 𝐴)
5920, 6, 21, 3leat2 34899 . . 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)))
602, 4, 58, 27, 44, 59syl32anc 1374 . 2 (𝜑𝐶 = ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)))
6160, 58eqeltrd 2730 1 (𝜑𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  Posetcpo 16987  ltcplt 16988  joincjn 16991  meetcmee 16992  0.cp0 17084  Latclat 17092  OPcops 34777  Atomscatm 34868  HLchlt 34955  LLinesclln 35095  LPlanesclpl 35096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103
This theorem is referenced by:  dalem2  35265  dalem5  35271  dalem-cly  35275  dalem9  35276  dalem19  35286  dalem21  35298  dalem25  35302
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