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Theorem dalem60 39237
Description: Lemma for dath 39241. 𝐡 is an axis of perspectivity (almost). (Contributed by NM, 11-Aug-2012.)
Hypotheses
Ref Expression
dalem.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)))))
dalem.l ≀ = (leβ€˜πΎ)
dalem.j ∨ = (joinβ€˜πΎ)
dalem.a 𝐴 = (Atomsβ€˜πΎ)
dalem.ps (πœ“ ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ Β¬ 𝑐 ≀ π‘Œ ∧ (𝑑 β‰  𝑐 ∧ Β¬ 𝑑 ≀ π‘Œ ∧ 𝐢 ≀ (𝑐 ∨ 𝑑))))
dalem60.m ∧ = (meetβ€˜πΎ)
dalem60.o 𝑂 = (LPlanesβ€˜πΎ)
dalem60.y π‘Œ = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem60.z 𝑍 = ((𝑆 ∨ 𝑇) ∨ π‘ˆ)
dalem60.d 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))
dalem60.e 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ))
dalem60.g 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
dalem60.h 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
dalem60.i 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ π‘ˆ))
dalem60.b1 𝐡 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ π‘Œ)
Assertion
Ref Expression
dalem60 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (𝐷 ∨ 𝐸) = 𝐡)

Proof of Theorem dalem60
StepHypRef Expression
1 dalem.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 dalem.l . . . 4 ≀ = (leβ€˜πΎ)
3 dalem.j . . . 4 ∨ = (joinβ€˜πΎ)
4 dalem.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
5 dalem.ps . . . 4 (πœ“ ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ Β¬ 𝑐 ≀ π‘Œ ∧ (𝑑 β‰  𝑐 ∧ Β¬ 𝑑 ≀ π‘Œ ∧ 𝐢 ≀ (𝑐 ∨ 𝑑))))
6 dalem60.m . . . 4 ∧ = (meetβ€˜πΎ)
7 dalem60.o . . . 4 𝑂 = (LPlanesβ€˜πΎ)
8 dalem60.y . . . 4 π‘Œ = ((𝑃 ∨ 𝑄) ∨ 𝑅)
9 dalem60.z . . . 4 𝑍 = ((𝑆 ∨ 𝑇) ∨ π‘ˆ)
10 dalem60.d . . . 4 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))
11 dalem60.g . . . 4 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
12 dalem60.h . . . 4 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
13 dalem60.i . . . 4 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ π‘ˆ))
14 dalem60.b1 . . . 4 𝐡 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ π‘Œ)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem57 39234 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐷 ≀ 𝐡)
16 dalem60.e . . . 4 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ))
171, 2, 3, 4, 5, 6, 7, 8, 9, 16, 11, 12, 13, 14dalem58 39235 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐸 ≀ 𝐡)
181dalemkelat 39129 . . . . 5 (πœ‘ β†’ 𝐾 ∈ Lat)
19183ad2ant1 1130 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐾 ∈ Lat)
201, 2, 3, 4, 6, 7, 8, 9, 10dalemdea 39167 . . . . . 6 (πœ‘ β†’ 𝐷 ∈ 𝐴)
21 eqid 2728 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2221, 4atbase 38793 . . . . . 6 (𝐷 ∈ 𝐴 β†’ 𝐷 ∈ (Baseβ€˜πΎ))
2320, 22syl 17 . . . . 5 (πœ‘ β†’ 𝐷 ∈ (Baseβ€˜πΎ))
24233ad2ant1 1130 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐷 ∈ (Baseβ€˜πΎ))
251, 2, 3, 4, 6, 7, 8, 9, 16dalemeea 39168 . . . . . 6 (πœ‘ β†’ 𝐸 ∈ 𝐴)
2621, 4atbase 38793 . . . . . 6 (𝐸 ∈ 𝐴 β†’ 𝐸 ∈ (Baseβ€˜πΎ))
2725, 26syl 17 . . . . 5 (πœ‘ β†’ 𝐸 ∈ (Baseβ€˜πΎ))
28273ad2ant1 1130 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐸 ∈ (Baseβ€˜πΎ))
29 eqid 2728 . . . . . 6 (LLinesβ€˜πΎ) = (LLinesβ€˜πΎ)
301, 2, 3, 4, 5, 6, 29, 7, 8, 9, 11, 12, 13, 14dalem53 39230 . . . . 5 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐡 ∈ (LLinesβ€˜πΎ))
3121, 29llnbase 39014 . . . . 5 (𝐡 ∈ (LLinesβ€˜πΎ) β†’ 𝐡 ∈ (Baseβ€˜πΎ))
3230, 31syl 17 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐡 ∈ (Baseβ€˜πΎ))
3321, 2, 3latjle12 18449 . . . 4 ((𝐾 ∈ Lat ∧ (𝐷 ∈ (Baseβ€˜πΎ) ∧ 𝐸 ∈ (Baseβ€˜πΎ) ∧ 𝐡 ∈ (Baseβ€˜πΎ))) β†’ ((𝐷 ≀ 𝐡 ∧ 𝐸 ≀ 𝐡) ↔ (𝐷 ∨ 𝐸) ≀ 𝐡))
3419, 24, 28, 32, 33syl13anc 1369 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐷 ≀ 𝐡 ∧ 𝐸 ≀ 𝐡) ↔ (𝐷 ∨ 𝐸) ≀ 𝐡))
3515, 17, 34mpbi2and 710 . 2 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (𝐷 ∨ 𝐸) ≀ 𝐡)
361dalemkehl 39128 . . . 4 (πœ‘ β†’ 𝐾 ∈ HL)
37363ad2ant1 1130 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐾 ∈ HL)
381, 2, 3, 4, 6, 7, 8, 9, 10, 16dalemdnee 39171 . . . . 5 (πœ‘ β†’ 𝐷 β‰  𝐸)
393, 4, 29llni2 39017 . . . . 5 (((𝐾 ∈ HL ∧ 𝐷 ∈ 𝐴 ∧ 𝐸 ∈ 𝐴) ∧ 𝐷 β‰  𝐸) β†’ (𝐷 ∨ 𝐸) ∈ (LLinesβ€˜πΎ))
4036, 20, 25, 38, 39syl31anc 1370 . . . 4 (πœ‘ β†’ (𝐷 ∨ 𝐸) ∈ (LLinesβ€˜πΎ))
41403ad2ant1 1130 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (𝐷 ∨ 𝐸) ∈ (LLinesβ€˜πΎ))
422, 29llncmp 39027 . . 3 ((𝐾 ∈ HL ∧ (𝐷 ∨ 𝐸) ∈ (LLinesβ€˜πΎ) ∧ 𝐡 ∈ (LLinesβ€˜πΎ)) β†’ ((𝐷 ∨ 𝐸) ≀ 𝐡 ↔ (𝐷 ∨ 𝐸) = 𝐡))
4337, 41, 30, 42syl3anc 1368 . 2 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐷 ∨ 𝐸) ≀ 𝐡 ↔ (𝐷 ∨ 𝐸) = 𝐡))
4435, 43mpbid 231 1 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (𝐷 ∨ 𝐸) = 𝐡)
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:  dalem61  39238
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