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Theorem dalemcea 38617
Description: Lemma for dath 38693. Frequently-used utility lemma. Here we show that 𝐢 must be an atom. This is an assumption in most presentations of Desargues's theorem; instead, we assume only the 𝐢 is a lattice element, in order to make later substitutions for 𝐢 easier. (Contributed by NM, 23-Sep-2012.)
Hypotheses
Ref Expression
dalema.ph (πœ‘ ↔ (((𝐾 ∈ HL ∧ 𝐢 ∈ (Baseβ€˜πΎ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) ∧ (π‘Œ ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((Β¬ 𝐢 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝐢 ≀ (𝑄 ∨ 𝑅) ∧ Β¬ 𝐢 ≀ (𝑅 ∨ 𝑃)) ∧ (Β¬ 𝐢 ≀ (𝑆 ∨ 𝑇) ∧ Β¬ 𝐢 ≀ (𝑇 ∨ π‘ˆ) ∧ Β¬ 𝐢 ≀ (π‘ˆ ∨ 𝑆)) ∧ (𝐢 ≀ (𝑃 ∨ 𝑆) ∧ 𝐢 ≀ (𝑄 ∨ 𝑇) ∧ 𝐢 ≀ (𝑅 ∨ π‘ˆ)))))
dalemc.l ≀ = (leβ€˜πΎ)
dalemc.j ∨ = (joinβ€˜πΎ)
dalemc.a 𝐴 = (Atomsβ€˜πΎ)
dalem1.o 𝑂 = (LPlanesβ€˜πΎ)
dalem1.y π‘Œ = ((𝑃 ∨ 𝑄) ∨ 𝑅)
Assertion
Ref Expression
dalemcea (πœ‘ β†’ 𝐢 ∈ 𝐴)

Proof of Theorem dalemcea
StepHypRef Expression
1 dalema.ph . . . 4 (πœ‘ ↔ (((𝐾 ∈ HL ∧ 𝐢 ∈ (Baseβ€˜πΎ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) ∧ (π‘Œ ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((Β¬ 𝐢 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝐢 ≀ (𝑄 ∨ 𝑅) ∧ Β¬ 𝐢 ≀ (𝑅 ∨ 𝑃)) ∧ (Β¬ 𝐢 ≀ (𝑆 ∨ 𝑇) ∧ Β¬ 𝐢 ≀ (𝑇 ∨ π‘ˆ) ∧ Β¬ 𝐢 ≀ (π‘ˆ ∨ 𝑆)) ∧ (𝐢 ≀ (𝑃 ∨ 𝑆) ∧ 𝐢 ≀ (𝑄 ∨ 𝑇) ∧ 𝐢 ≀ (𝑅 ∨ π‘ˆ)))))
21dalemkeop 38582 . . 3 (πœ‘ β†’ 𝐾 ∈ OP)
3 dalemc.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
41, 3dalemceb 38595 . . 3 (πœ‘ β†’ 𝐢 ∈ (Baseβ€˜πΎ))
51dalemkehl 38580 . . . 4 (πœ‘ β†’ 𝐾 ∈ HL)
6 dalemc.l . . . . 5 ≀ = (leβ€˜πΎ)
7 dalemc.j . . . . 5 ∨ = (joinβ€˜πΎ)
8 dalem1.o . . . . 5 𝑂 = (LPlanesβ€˜πΎ)
9 dalem1.y . . . . 5 π‘Œ = ((𝑃 ∨ 𝑄) ∨ 𝑅)
101, 6, 7, 3, 8, 9dalempjsen 38610 . . . 4 (πœ‘ β†’ (𝑃 ∨ 𝑆) ∈ (LLinesβ€˜πΎ))
111dalemqea 38584 . . . . 5 (πœ‘ β†’ 𝑄 ∈ 𝐴)
121dalemtea 38587 . . . . 5 (πœ‘ β†’ 𝑇 ∈ 𝐴)
131, 6, 7, 3, 8, 9dalemqnet 38609 . . . . 5 (πœ‘ β†’ 𝑄 β‰  𝑇)
14 eqid 2732 . . . . . 6 (LLinesβ€˜πΎ) = (LLinesβ€˜πΎ)
157, 3, 14llni2 38469 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑄 β‰  𝑇) β†’ (𝑄 ∨ 𝑇) ∈ (LLinesβ€˜πΎ))
165, 11, 12, 13, 15syl31anc 1373 . . . 4 (πœ‘ β†’ (𝑄 ∨ 𝑇) ∈ (LLinesβ€˜πΎ))
171, 6, 7, 3, 8, 9dalem1 38616 . . . 4 (πœ‘ β†’ (𝑃 ∨ 𝑆) β‰  (𝑄 ∨ 𝑇))
181dalem-clpjq 38594 . . . . . . . 8 (πœ‘ β†’ Β¬ 𝐢 ≀ (𝑃 ∨ 𝑄))
191, 7, 3dalempjqeb 38602 . . . . . . . . . . 11 (πœ‘ β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
20 eqid 2732 . . . . . . . . . . . 12 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
21 eqid 2732 . . . . . . . . . . . 12 (0.β€˜πΎ) = (0.β€˜πΎ)
2220, 6, 21op0le 38142 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ)) β†’ (0.β€˜πΎ) ≀ (𝑃 ∨ 𝑄))
232, 19, 22syl2anc 584 . . . . . . . . . 10 (πœ‘ β†’ (0.β€˜πΎ) ≀ (𝑃 ∨ 𝑄))
24 breq1 5151 . . . . . . . . . 10 (𝐢 = (0.β€˜πΎ) β†’ (𝐢 ≀ (𝑃 ∨ 𝑄) ↔ (0.β€˜πΎ) ≀ (𝑃 ∨ 𝑄)))
2523, 24syl5ibrcom 246 . . . . . . . . 9 (πœ‘ β†’ (𝐢 = (0.β€˜πΎ) β†’ 𝐢 ≀ (𝑃 ∨ 𝑄)))
2625necon3bd 2954 . . . . . . . 8 (πœ‘ β†’ (Β¬ 𝐢 ≀ (𝑃 ∨ 𝑄) β†’ 𝐢 β‰  (0.β€˜πΎ)))
2718, 26mpd 15 . . . . . . 7 (πœ‘ β†’ 𝐢 β‰  (0.β€˜πΎ))
28 eqid 2732 . . . . . . . . 9 (ltβ€˜πΎ) = (ltβ€˜πΎ)
2920, 28, 21opltn0 38146 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝐢 ∈ (Baseβ€˜πΎ)) β†’ ((0.β€˜πΎ)(ltβ€˜πΎ)𝐢 ↔ 𝐢 β‰  (0.β€˜πΎ)))
302, 4, 29syl2anc 584 . . . . . . 7 (πœ‘ β†’ ((0.β€˜πΎ)(ltβ€˜πΎ)𝐢 ↔ 𝐢 β‰  (0.β€˜πΎ)))
3127, 30mpbird 256 . . . . . 6 (πœ‘ β†’ (0.β€˜πΎ)(ltβ€˜πΎ)𝐢)
321dalemclpjs 38591 . . . . . . 7 (πœ‘ β†’ 𝐢 ≀ (𝑃 ∨ 𝑆))
331dalemclqjt 38592 . . . . . . 7 (πœ‘ β†’ 𝐢 ≀ (𝑄 ∨ 𝑇))
341dalemkelat 38581 . . . . . . . 8 (πœ‘ β†’ 𝐾 ∈ Lat)
351dalempea 38583 . . . . . . . . 9 (πœ‘ β†’ 𝑃 ∈ 𝐴)
361dalemsea 38586 . . . . . . . . 9 (πœ‘ β†’ 𝑆 ∈ 𝐴)
3720, 7, 3hlatjcl 38323 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
385, 35, 36, 37syl3anc 1371 . . . . . . . 8 (πœ‘ β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
3920, 7, 3hlatjcl 38323 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) β†’ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
405, 11, 12, 39syl3anc 1371 . . . . . . . 8 (πœ‘ β†’ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
41 eqid 2732 . . . . . . . . 9 (meetβ€˜πΎ) = (meetβ€˜πΎ)
4220, 6, 41latlem12 18421 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐢 ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ) ∧ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ))) β†’ ((𝐢 ≀ (𝑃 ∨ 𝑆) ∧ 𝐢 ≀ (𝑄 ∨ 𝑇)) ↔ 𝐢 ≀ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇))))
4334, 4, 38, 40, 42syl13anc 1372 . . . . . . 7 (πœ‘ β†’ ((𝐢 ≀ (𝑃 ∨ 𝑆) ∧ 𝐢 ≀ (𝑄 ∨ 𝑇)) ↔ 𝐢 ≀ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇))))
4432, 33, 43mpbi2and 710 . . . . . 6 (πœ‘ β†’ 𝐢 ≀ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)))
45 opposet 38137 . . . . . . . 8 (𝐾 ∈ OP β†’ 𝐾 ∈ Poset)
462, 45syl 17 . . . . . . 7 (πœ‘ β†’ 𝐾 ∈ Poset)
4720, 21op0cl 38140 . . . . . . . 8 (𝐾 ∈ OP β†’ (0.β€˜πΎ) ∈ (Baseβ€˜πΎ))
482, 47syl 17 . . . . . . 7 (πœ‘ β†’ (0.β€˜πΎ) ∈ (Baseβ€˜πΎ))
4920, 41latmcl 18395 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ) ∧ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) ∈ (Baseβ€˜πΎ))
5034, 38, 40, 49syl3anc 1371 . . . . . . 7 (πœ‘ β†’ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) ∈ (Baseβ€˜πΎ))
5120, 6, 28pltletr 18298 . . . . . . 7 ((𝐾 ∈ Poset ∧ ((0.β€˜πΎ) ∈ (Baseβ€˜πΎ) ∧ 𝐢 ∈ (Baseβ€˜πΎ) ∧ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) ∈ (Baseβ€˜πΎ))) β†’ (((0.β€˜πΎ)(ltβ€˜πΎ)𝐢 ∧ 𝐢 ≀ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇))) β†’ (0.β€˜πΎ)(ltβ€˜πΎ)((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇))))
5246, 48, 4, 50, 51syl13anc 1372 . . . . . 6 (πœ‘ β†’ (((0.β€˜πΎ)(ltβ€˜πΎ)𝐢 ∧ 𝐢 ≀ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇))) β†’ (0.β€˜πΎ)(ltβ€˜πΎ)((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇))))
5331, 44, 52mp2and 697 . . . . 5 (πœ‘ β†’ (0.β€˜πΎ)(ltβ€˜πΎ)((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)))
5420, 28, 21opltn0 38146 . . . . . 6 ((𝐾 ∈ OP ∧ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) ∈ (Baseβ€˜πΎ)) β†’ ((0.β€˜πΎ)(ltβ€˜πΎ)((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) ↔ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) β‰  (0.β€˜πΎ)))
552, 50, 54syl2anc 584 . . . . 5 (πœ‘ β†’ ((0.β€˜πΎ)(ltβ€˜πΎ)((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) ↔ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) β‰  (0.β€˜πΎ)))
5653, 55mpbid 231 . . . 4 (πœ‘ β†’ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) β‰  (0.β€˜πΎ))
5741, 21, 3, 142llnmat 38481 . . . 4 (((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑆) ∈ (LLinesβ€˜πΎ) ∧ (𝑄 ∨ 𝑇) ∈ (LLinesβ€˜πΎ)) ∧ ((𝑃 ∨ 𝑆) β‰  (𝑄 ∨ 𝑇) ∧ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) β‰  (0.β€˜πΎ))) β†’ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) ∈ 𝐴)
585, 10, 16, 17, 56, 57syl32anc 1378 . . 3 (πœ‘ β†’ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) ∈ 𝐴)
5920, 6, 21, 3leat2 38250 . . 3 (((𝐾 ∈ OP ∧ 𝐢 ∈ (Baseβ€˜πΎ) ∧ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) ∈ 𝐴) ∧ (𝐢 β‰  (0.β€˜πΎ) ∧ 𝐢 ≀ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)))) β†’ 𝐢 = ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)))
602, 4, 58, 27, 44, 59syl32anc 1378 . 2 (πœ‘ β†’ 𝐢 = ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)))
6160, 58eqeltrd 2833 1 (πœ‘ β†’ 𝐢 ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  lecple 17206  Posetcpo 18262  ltcplt 18263  joincjn 18266  meetcmee 18267  0.cp0 18378  Latclat 18386  OPcops 38128  Atomscatm 38219  HLchlt 38306  LLinesclln 38448  LPlanesclpl 38449
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-proset 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-lat 18387  df-clat 18454  df-oposet 38132  df-ol 38134  df-oml 38135  df-covers 38222  df-ats 38223  df-atl 38254  df-cvlat 38278  df-hlat 38307  df-llines 38455  df-lplanes 38456
This theorem is referenced by:  dalem2  38618  dalem5  38624  dalem-cly  38628  dalem9  38629  dalem19  38639  dalem21  38651  dalem25  38655
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