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Theorem dalemcea 35438
Description: Lemma for dath 35514. 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)) ∧ (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 35403 . . 3 (𝜑𝐾 ∈ OP)
3 dalemc.a . . . 4 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 35416 . . 3 (𝜑𝐶 ∈ (Base‘𝐾))
51dalemkehl 35401 . . . 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 35431 . . . 4 (𝜑 → (𝑃 𝑆) ∈ (LLines‘𝐾))
111dalemqea 35405 . . . . 5 (𝜑𝑄𝐴)
121dalemtea 35408 . . . . 5 (𝜑𝑇𝐴)
131, 6, 7, 3, 8, 9dalemqnet 35430 . . . . 5 (𝜑𝑄𝑇)
14 eqid 2804 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
157, 3, 14llni2 35290 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) ∧ 𝑄𝑇) → (𝑄 𝑇) ∈ (LLines‘𝐾))
165, 11, 12, 13, 15syl31anc 1485 . . . 4 (𝜑 → (𝑄 𝑇) ∈ (LLines‘𝐾))
171, 6, 7, 3, 8, 9dalem1 35437 . . . 4 (𝜑 → (𝑃 𝑆) ≠ (𝑄 𝑇))
181dalem-clpjq 35415 . . . . . . . 8 (𝜑 → ¬ 𝐶 (𝑃 𝑄))
191, 7, 3dalempjqeb 35423 . . . . . . . . . . 11 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
20 eqid 2804 . . . . . . . . . . . 12 (Base‘𝐾) = (Base‘𝐾)
21 eqid 2804 . . . . . . . . . . . 12 (0.‘𝐾) = (0.‘𝐾)
2220, 6, 21op0le 34964 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (0.‘𝐾) (𝑃 𝑄))
232, 19, 22syl2anc 575 . . . . . . . . . 10 (𝜑 → (0.‘𝐾) (𝑃 𝑄))
24 breq1 4845 . . . . . . . . . 10 (𝐶 = (0.‘𝐾) → (𝐶 (𝑃 𝑄) ↔ (0.‘𝐾) (𝑃 𝑄)))
2523, 24syl5ibrcom 238 . . . . . . . . 9 (𝜑 → (𝐶 = (0.‘𝐾) → 𝐶 (𝑃 𝑄)))
2625necon3bd 2990 . . . . . . . 8 (𝜑 → (¬ 𝐶 (𝑃 𝑄) → 𝐶 ≠ (0.‘𝐾)))
2718, 26mpd 15 . . . . . . 7 (𝜑𝐶 ≠ (0.‘𝐾))
28 eqid 2804 . . . . . . . . 9 (lt‘𝐾) = (lt‘𝐾)
2920, 28, 21opltn0 34968 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝐶 ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)𝐶𝐶 ≠ (0.‘𝐾)))
302, 4, 29syl2anc 575 . . . . . . 7 (𝜑 → ((0.‘𝐾)(lt‘𝐾)𝐶𝐶 ≠ (0.‘𝐾)))
3127, 30mpbird 248 . . . . . 6 (𝜑 → (0.‘𝐾)(lt‘𝐾)𝐶)
321dalemclpjs 35412 . . . . . . 7 (𝜑𝐶 (𝑃 𝑆))
331dalemclqjt 35413 . . . . . . 7 (𝜑𝐶 (𝑄 𝑇))
341dalemkelat 35402 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
351dalempea 35404 . . . . . . . . 9 (𝜑𝑃𝐴)
361dalemsea 35407 . . . . . . . . 9 (𝜑𝑆𝐴)
3720, 7, 3hlatjcl 35145 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
385, 35, 36, 37syl3anc 1483 . . . . . . . 8 (𝜑 → (𝑃 𝑆) ∈ (Base‘𝐾))
3920, 7, 3hlatjcl 35145 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) → (𝑄 𝑇) ∈ (Base‘𝐾))
405, 11, 12, 39syl3anc 1483 . . . . . . . 8 (𝜑 → (𝑄 𝑇) ∈ (Base‘𝐾))
41 eqid 2804 . . . . . . . . 9 (meet‘𝐾) = (meet‘𝐾)
4220, 6, 41latlem12 17281 . . . . . . . 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 1484 . . . . . . 7 (𝜑 → ((𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇)) ↔ 𝐶 ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇))))
4432, 33, 43mpbi2and 694 . . . . . 6 (𝜑𝐶 ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)))
45 opposet 34959 . . . . . . . 8 (𝐾 ∈ OP → 𝐾 ∈ Poset)
462, 45syl 17 . . . . . . 7 (𝜑𝐾 ∈ Poset)
4720, 21op0cl 34962 . . . . . . . 8 (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
482, 47syl 17 . . . . . . 7 (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
4920, 41latmcl 17255 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ (Base‘𝐾))
5034, 38, 40, 49syl3anc 1483 . . . . . . 7 (𝜑 → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ (Base‘𝐾))
5120, 6, 28pltletr 17174 . . . . . . 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 1484 . . . . . 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 682 . . . . 5 (𝜑 → (0.‘𝐾)(lt‘𝐾)((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)))
5420, 28, 21opltn0 34968 . . . . . 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 575 . . . . 5 (𝜑 → ((0.‘𝐾)(lt‘𝐾)((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ↔ ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ≠ (0.‘𝐾)))
5653, 55mpbid 223 . . . 4 (𝜑 → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ≠ (0.‘𝐾))
5741, 21, 3, 142llnmat 35302 . . . 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 1490 . . 3 (𝜑 → ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)) ∈ 𝐴)
5920, 6, 21, 3leat2 35072 . . 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 1490 . 2 (𝜑𝐶 = ((𝑃 𝑆)(meet‘𝐾)(𝑄 𝑇)))
6160, 58eqeltrd 2883 1 (𝜑𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wne 2976   class class class wbr 4842  cfv 6099  (class class class)co 6872  Basecbs 16066  lecple 16158  Posetcpo 17143  ltcplt 17144  joincjn 17147  meetcmee 17148  0.cp0 17240  Latclat 17248  OPcops 34950  Atomscatm 35041  HLchlt 35128  LLinesclln 35269  LPlanesclpl 35270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5094  ax-un 7177
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ne 2977  df-ral 3099  df-rex 3100  df-reu 3101  df-rab 3103  df-v 3391  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4115  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5217  df-xp 5315  df-rel 5316  df-cnv 5317  df-co 5318  df-dm 5319  df-rn 5320  df-res 5321  df-ima 5322  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6833  df-ov 6875  df-oprab 6876  df-proset 17131  df-poset 17149  df-plt 17161  df-lub 17177  df-glb 17178  df-join 17179  df-meet 17180  df-p0 17242  df-lat 17249  df-clat 17311  df-oposet 34954  df-ol 34956  df-oml 34957  df-covers 35044  df-ats 35045  df-atl 35076  df-cvlat 35100  df-hlat 35129  df-llines 35276  df-lplanes 35277
This theorem is referenced by:  dalem2  35439  dalem5  35445  dalem-cly  35449  dalem9  35450  dalem19  35460  dalem21  35472  dalem25  35476
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