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Theorem dalemcea 38531
Description: Lemma for dath 38607. 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 38496 . . 3 (πœ‘ β†’ 𝐾 ∈ OP)
3 dalemc.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
41, 3dalemceb 38509 . . 3 (πœ‘ β†’ 𝐢 ∈ (Baseβ€˜πΎ))
51dalemkehl 38494 . . . 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 38524 . . . 4 (πœ‘ β†’ (𝑃 ∨ 𝑆) ∈ (LLinesβ€˜πΎ))
111dalemqea 38498 . . . . 5 (πœ‘ β†’ 𝑄 ∈ 𝐴)
121dalemtea 38501 . . . . 5 (πœ‘ β†’ 𝑇 ∈ 𝐴)
131, 6, 7, 3, 8, 9dalemqnet 38523 . . . . 5 (πœ‘ β†’ 𝑄 β‰  𝑇)
14 eqid 2733 . . . . . 6 (LLinesβ€˜πΎ) = (LLinesβ€˜πΎ)
157, 3, 14llni2 38383 . . . . 5 (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑄 β‰  𝑇) β†’ (𝑄 ∨ 𝑇) ∈ (LLinesβ€˜πΎ))
165, 11, 12, 13, 15syl31anc 1374 . . . 4 (πœ‘ β†’ (𝑄 ∨ 𝑇) ∈ (LLinesβ€˜πΎ))
171, 6, 7, 3, 8, 9dalem1 38530 . . . 4 (πœ‘ β†’ (𝑃 ∨ 𝑆) β‰  (𝑄 ∨ 𝑇))
181dalem-clpjq 38508 . . . . . . . 8 (πœ‘ β†’ Β¬ 𝐢 ≀ (𝑃 ∨ 𝑄))
191, 7, 3dalempjqeb 38516 . . . . . . . . . . 11 (πœ‘ β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
20 eqid 2733 . . . . . . . . . . . 12 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
21 eqid 2733 . . . . . . . . . . . 12 (0.β€˜πΎ) = (0.β€˜πΎ)
2220, 6, 21op0le 38056 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ)) β†’ (0.β€˜πΎ) ≀ (𝑃 ∨ 𝑄))
232, 19, 22syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ (0.β€˜πΎ) ≀ (𝑃 ∨ 𝑄))
24 breq1 5152 . . . . . . . . . 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 2955 . . . . . . . 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 2733 . . . . . . . . 9 (ltβ€˜πΎ) = (ltβ€˜πΎ)
2920, 28, 21opltn0 38060 . . . . . . . 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 585 . . . . . . 7 (πœ‘ β†’ ((0.β€˜πΎ)(ltβ€˜πΎ)𝐢 ↔ 𝐢 β‰  (0.β€˜πΎ)))
3127, 30mpbird 257 . . . . . 6 (πœ‘ β†’ (0.β€˜πΎ)(ltβ€˜πΎ)𝐢)
321dalemclpjs 38505 . . . . . . 7 (πœ‘ β†’ 𝐢 ≀ (𝑃 ∨ 𝑆))
331dalemclqjt 38506 . . . . . . 7 (πœ‘ β†’ 𝐢 ≀ (𝑄 ∨ 𝑇))
341dalemkelat 38495 . . . . . . . 8 (πœ‘ β†’ 𝐾 ∈ Lat)
351dalempea 38497 . . . . . . . . 9 (πœ‘ β†’ 𝑃 ∈ 𝐴)
361dalemsea 38500 . . . . . . . . 9 (πœ‘ β†’ 𝑆 ∈ 𝐴)
3720, 7, 3hlatjcl 38237 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
385, 35, 36, 37syl3anc 1372 . . . . . . . 8 (πœ‘ β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
3920, 7, 3hlatjcl 38237 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) β†’ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
405, 11, 12, 39syl3anc 1372 . . . . . . . 8 (πœ‘ β†’ (𝑄 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
41 eqid 2733 . . . . . . . . 9 (meetβ€˜πΎ) = (meetβ€˜πΎ)
4220, 6, 41latlem12 18419 . . . . . . . 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 1373 . . . . . . 7 (πœ‘ β†’ ((𝐢 ≀ (𝑃 ∨ 𝑆) ∧ 𝐢 ≀ (𝑄 ∨ 𝑇)) ↔ 𝐢 ≀ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇))))
4432, 33, 43mpbi2and 711 . . . . . 6 (πœ‘ β†’ 𝐢 ≀ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)))
45 opposet 38051 . . . . . . . 8 (𝐾 ∈ OP β†’ 𝐾 ∈ Poset)
462, 45syl 17 . . . . . . 7 (πœ‘ β†’ 𝐾 ∈ Poset)
4720, 21op0cl 38054 . . . . . . . 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 18393 . . . . . . . 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 1372 . . . . . . 7 (πœ‘ β†’ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) ∈ (Baseβ€˜πΎ))
5120, 6, 28pltletr 18296 . . . . . . 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 1373 . . . . . 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 698 . . . . 5 (πœ‘ β†’ (0.β€˜πΎ)(ltβ€˜πΎ)((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)))
5420, 28, 21opltn0 38060 . . . . . 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 585 . . . . 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 38395 . . . 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 1379 . . 3 (πœ‘ β†’ ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)) ∈ 𝐴)
5920, 6, 21, 3leat2 38164 . . 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 1379 . 2 (πœ‘ β†’ 𝐢 = ((𝑃 ∨ 𝑆)(meetβ€˜πΎ)(𝑄 ∨ 𝑇)))
6160, 58eqeltrd 2834 1 (πœ‘ β†’ 𝐢 ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  Posetcpo 18260  ltcplt 18261  joincjn 18264  meetcmee 18265  0.cp0 18376  Latclat 18384  OPcops 38042  Atomscatm 38133  HLchlt 38220  LLinesclln 38362  LPlanesclpl 38363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-llines 38369  df-lplanes 38370
This theorem is referenced by:  dalem2  38532  dalem5  38538  dalem-cly  38542  dalem9  38543  dalem19  38553  dalem21  38565  dalem25  38569
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