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Theorem dalem16 39184
Description: Lemma for dath 39241. The atoms 𝐷, 𝐸, and 𝐹 form a line of perspectivity. This is Desargues's theorem for the special case where planes π‘Œ and 𝑍 are different. (Contributed by NM, 7-Aug-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)
dalem16.m ∧ = (meetβ€˜πΎ)
dalem16.o 𝑂 = (LPlanesβ€˜πΎ)
dalem16.y π‘Œ = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem16.z 𝑍 = ((𝑆 ∨ 𝑇) ∨ π‘ˆ)
dalem16.d 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))
dalem16.e 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ))
dalem16.f 𝐹 = ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))
Assertion
Ref Expression
dalem16 ((πœ‘ ∧ π‘Œ β‰  𝑍) β†’ 𝐹 ≀ (𝐷 ∨ 𝐸))

Proof of Theorem dalem16
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)))))
2 dalemc.l . . . 4 ≀ = (leβ€˜πΎ)
3 dalemc.j . . . 4 ∨ = (joinβ€˜πΎ)
4 dalemc.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
5 dalem16.m . . . 4 ∧ = (meetβ€˜πΎ)
6 dalem16.o . . . 4 𝑂 = (LPlanesβ€˜πΎ)
7 dalem16.y . . . 4 π‘Œ = ((𝑃 ∨ 𝑄) ∨ 𝑅)
8 dalem16.z . . . 4 𝑍 = ((𝑆 ∨ 𝑇) ∨ π‘ˆ)
9 eqid 2728 . . . 4 (π‘Œ ∧ 𝑍) = (π‘Œ ∧ 𝑍)
10 dalem16.f . . . 4 𝐹 = ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem12 39180 . . 3 (πœ‘ β†’ 𝐹 ≀ (π‘Œ ∧ 𝑍))
1211adantr 479 . 2 ((πœ‘ ∧ π‘Œ β‰  𝑍) β†’ 𝐹 ≀ (π‘Œ ∧ 𝑍))
13 dalem16.d . . . . . 6 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))
141, 2, 3, 4, 5, 6, 7, 8, 9, 13dalem10 39178 . . . . 5 (πœ‘ β†’ 𝐷 ≀ (π‘Œ ∧ 𝑍))
15 dalem16.e . . . . . 6 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ))
161, 2, 3, 4, 5, 6, 7, 8, 9, 15dalem11 39179 . . . . 5 (πœ‘ β†’ 𝐸 ≀ (π‘Œ ∧ 𝑍))
171dalemkelat 39129 . . . . . 6 (πœ‘ β†’ 𝐾 ∈ Lat)
181, 2, 3, 4, 5, 6, 7, 8, 13dalemdea 39167 . . . . . . 7 (πœ‘ β†’ 𝐷 ∈ 𝐴)
19 eqid 2728 . . . . . . . 8 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2019, 4atbase 38793 . . . . . . 7 (𝐷 ∈ 𝐴 β†’ 𝐷 ∈ (Baseβ€˜πΎ))
2118, 20syl 17 . . . . . 6 (πœ‘ β†’ 𝐷 ∈ (Baseβ€˜πΎ))
221, 2, 3, 4, 5, 6, 7, 8, 15dalemeea 39168 . . . . . . 7 (πœ‘ β†’ 𝐸 ∈ 𝐴)
2319, 4atbase 38793 . . . . . . 7 (𝐸 ∈ 𝐴 β†’ 𝐸 ∈ (Baseβ€˜πΎ))
2422, 23syl 17 . . . . . 6 (πœ‘ β†’ 𝐸 ∈ (Baseβ€˜πΎ))
251, 6dalemyeb 39154 . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ (Baseβ€˜πΎ))
261dalemzeo 39138 . . . . . . . 8 (πœ‘ β†’ 𝑍 ∈ 𝑂)
2719, 6lplnbase 39039 . . . . . . . 8 (𝑍 ∈ 𝑂 β†’ 𝑍 ∈ (Baseβ€˜πΎ))
2826, 27syl 17 . . . . . . 7 (πœ‘ β†’ 𝑍 ∈ (Baseβ€˜πΎ))
2919, 5latmcl 18439 . . . . . . 7 ((𝐾 ∈ Lat ∧ π‘Œ ∈ (Baseβ€˜πΎ) ∧ 𝑍 ∈ (Baseβ€˜πΎ)) β†’ (π‘Œ ∧ 𝑍) ∈ (Baseβ€˜πΎ))
3017, 25, 28, 29syl3anc 1368 . . . . . 6 (πœ‘ β†’ (π‘Œ ∧ 𝑍) ∈ (Baseβ€˜πΎ))
3119, 2, 3latjle12 18449 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐷 ∈ (Baseβ€˜πΎ) ∧ 𝐸 ∈ (Baseβ€˜πΎ) ∧ (π‘Œ ∧ 𝑍) ∈ (Baseβ€˜πΎ))) β†’ ((𝐷 ≀ (π‘Œ ∧ 𝑍) ∧ 𝐸 ≀ (π‘Œ ∧ 𝑍)) ↔ (𝐷 ∨ 𝐸) ≀ (π‘Œ ∧ 𝑍)))
3217, 21, 24, 30, 31syl13anc 1369 . . . . 5 (πœ‘ β†’ ((𝐷 ≀ (π‘Œ ∧ 𝑍) ∧ 𝐸 ≀ (π‘Œ ∧ 𝑍)) ↔ (𝐷 ∨ 𝐸) ≀ (π‘Œ ∧ 𝑍)))
3314, 16, 32mpbi2and 710 . . . 4 (πœ‘ β†’ (𝐷 ∨ 𝐸) ≀ (π‘Œ ∧ 𝑍))
3433adantr 479 . . 3 ((πœ‘ ∧ π‘Œ β‰  𝑍) β†’ (𝐷 ∨ 𝐸) ≀ (π‘Œ ∧ 𝑍))
351dalemkehl 39128 . . . . 5 (πœ‘ β†’ 𝐾 ∈ HL)
3635adantr 479 . . . 4 ((πœ‘ ∧ π‘Œ β‰  𝑍) β†’ 𝐾 ∈ HL)
371, 2, 3, 4, 5, 6, 7, 8, 13, 15dalemdnee 39171 . . . . . 6 (πœ‘ β†’ 𝐷 β‰  𝐸)
38 eqid 2728 . . . . . . 7 (LLinesβ€˜πΎ) = (LLinesβ€˜πΎ)
393, 4, 38llni2 39017 . . . . . 6 (((𝐾 ∈ HL ∧ 𝐷 ∈ 𝐴 ∧ 𝐸 ∈ 𝐴) ∧ 𝐷 β‰  𝐸) β†’ (𝐷 ∨ 𝐸) ∈ (LLinesβ€˜πΎ))
4035, 18, 22, 37, 39syl31anc 1370 . . . . 5 (πœ‘ β†’ (𝐷 ∨ 𝐸) ∈ (LLinesβ€˜πΎ))
4140adantr 479 . . . 4 ((πœ‘ ∧ π‘Œ β‰  𝑍) β†’ (𝐷 ∨ 𝐸) ∈ (LLinesβ€˜πΎ))
421, 2, 3, 4, 5, 38, 6, 7, 8, 9dalem15 39183 . . . 4 ((πœ‘ ∧ π‘Œ β‰  𝑍) β†’ (π‘Œ ∧ 𝑍) ∈ (LLinesβ€˜πΎ))
432, 38llncmp 39027 . . . 4 ((𝐾 ∈ HL ∧ (𝐷 ∨ 𝐸) ∈ (LLinesβ€˜πΎ) ∧ (π‘Œ ∧ 𝑍) ∈ (LLinesβ€˜πΎ)) β†’ ((𝐷 ∨ 𝐸) ≀ (π‘Œ ∧ 𝑍) ↔ (𝐷 ∨ 𝐸) = (π‘Œ ∧ 𝑍)))
4436, 41, 42, 43syl3anc 1368 . . 3 ((πœ‘ ∧ π‘Œ β‰  𝑍) β†’ ((𝐷 ∨ 𝐸) ≀ (π‘Œ ∧ 𝑍) ↔ (𝐷 ∨ 𝐸) = (π‘Œ ∧ 𝑍)))
4534, 44mpbid 231 . 2 ((πœ‘ ∧ π‘Œ β‰  𝑍) β†’ (𝐷 ∨ 𝐸) = (π‘Œ ∧ 𝑍))
4612, 45breqtrrd 5180 1 ((πœ‘ ∧ π‘Œ β‰  𝑍) β†’ 𝐹 ≀ (𝐷 ∨ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2937   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  lecple 17247  joincjn 18310  meetcmee 18311  Latclat 18430  Atomscatm 38767  HLchlt 38854  LLinesclln 38996  LPlanesclpl 38997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-llines 39003  df-lplanes 39004  df-lvols 39005
This theorem is referenced by:  dalem63  39240
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