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Theorem dalemcea 35468
 Description: Lemma for dath 35544. 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 35433 . . 3 (𝜑𝐾 ∈ OP)
3 dalemc.a . . . 4 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 35446 . . 3 (𝜑𝐶 ∈ (Base‘𝐾))
51dalemkehl 35431 . . . 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 35461 . . . 4 (𝜑 → (𝑃 𝑆) ∈ (LLines‘𝐾))
111dalemqea 35435 . . . . 5 (𝜑𝑄𝐴)
121dalemtea 35438 . . . . 5 (𝜑𝑇𝐴)
131, 6, 7, 3, 8, 9dalemqnet 35460 . . . . 5 (𝜑𝑄𝑇)
14 eqid 2771 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
157, 3, 14llni2 35320 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) ∧ 𝑄𝑇) → (𝑄 𝑇) ∈ (LLines‘𝐾))
165, 11, 12, 13, 15syl31anc 1479 . . . 4 (𝜑 → (𝑄 𝑇) ∈ (LLines‘𝐾))
171, 6, 7, 3, 8, 9dalem1 35467 . . . 4 (𝜑 → (𝑃 𝑆) ≠ (𝑄 𝑇))
181dalem-clpjq 35445 . . . . . . . 8 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
191, 7, 3dalempjqeb 35453 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
20 eqid 2771 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
21 eqid 2771 . . . . . . . . . . . 12 (0.‘𝐾) = (0.‘𝐾)
2220, 6, 21op0le 34995 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (0.‘𝐾) (𝑃 𝑄))
232, 19, 22syl2anc 573 . . . . . . . . . 10 (𝜑 → (0.‘𝐾) (𝑃 𝑄))
24 breq1 4789 . . . . . . . . . 10 (𝐶 = (0.‘𝐾) → (𝐶 (𝑃 𝑄) ↔ (0.‘𝐾) (𝑃 𝑄)))
2523, 24syl5ibrcom 237 . . . . . . . . 9 (𝜑 → (𝐶 = (0.‘𝐾) → 𝐶 (𝑃 𝑄)))
2625necon3bd 2957 . . . . . . . 8 (𝜑 → (¬ 𝐶 (𝑃 𝑄) → 𝐶 ≠ (0.‘𝐾)))
2718, 26mpd 15 . . . . . . 7 (𝜑𝐶 ≠ (0.‘𝐾))
28 eqid 2771 . . . . . . . . 9 (lt‘𝐾) = (lt‘𝐾)
2920, 28, 21opltn0 34999 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝐶 ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)𝐶𝐶 ≠ (0.‘𝐾)))
302, 4, 29syl2anc 573 . . . . . . 7 (𝜑 → ((0.‘𝐾)(lt‘𝐾)𝐶𝐶 ≠ (0.‘𝐾)))
3127, 30mpbird 247 . . . . . 6 (𝜑 → (0.‘𝐾)(lt‘𝐾)𝐶)
321dalemclpjs 35442 . . . . . . 7 (𝜑𝐶 (𝑃 𝑆))
331dalemclqjt 35443 . . . . . . 7 (𝜑𝐶 (𝑄 𝑇))
341dalemkelat 35432 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
351dalempea 35434 . . . . . . . . 9 (𝜑𝑃𝐴)
361dalemsea 35437 . . . . . . . . 9 (𝜑𝑆𝐴)
3720, 7, 3hlatjcl 35175 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
385, 35, 36, 37syl3anc 1476 . . . . . . . 8 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
3920, 7, 3hlatjcl 35175 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) ∈ (Base‘𝐾))
405, 11, 12, 39syl3anc 1476 . . . . . . . 8 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
41 eqid 2771 . . . . . . . . 9 (meet‘𝐾) = (meet‘𝐾)
4220, 6, 41latlem12 17286 . . . . . . . 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 1478 . . . . . . 7 (𝜑 → ((𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇)) ↔ 𝐶 ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇))))
4432, 33, 43mpbi2and 691 . . . . . 6 (𝜑𝐶 ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)))
45 opposet 34990 . . . . . . . 8 (𝐾 ∈ OP → 𝐾 ∈ Poset)
462, 45syl 17 . . . . . . 7 (𝜑𝐾 ∈ Poset)
4720, 21op0cl 34993 . . . . . . . 8 (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
482, 47syl 17 . . . . . . 7 (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
4920, 41latmcl 17260 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ (Base‘𝐾))
5034, 38, 40, 49syl3anc 1476 . . . . . . 7 (𝜑 → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ (Base‘𝐾))
5120, 6, 28pltletr 17179 . . . . . . 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 1478 . . . . . 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 679 . . . . 5 (𝜑 → (0.‘𝐾)(lt‘𝐾)((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)))
5420, 28, 21opltn0 34999 . . . . . 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 573 . . . . 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 35332 . . . 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 1484 . . 3 (𝜑 → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ 𝐴)
5920, 6, 21, 3leat2 35103 . . 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 1484 . 2 (𝜑𝐶 = ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)))
6160, 58eqeltrd 2850 1 (𝜑𝐶𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ≠ wne 2943   class class class wbr 4786  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  Posetcpo 17148  ltcplt 17149  joincjn 17152  meetcmee 17153  0.cp0 17245  Latclat 17253  OPcops 34981  Atomscatm 35072  HLchlt 35159  LLinesclln 35299  LPlanesclpl 35300 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306  df-lplanes 35307 This theorem is referenced by:  dalem2  35469  dalem5  35475  dalem-cly  35479  dalem9  35480  dalem19  35490  dalem21  35502  dalem25  35506
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