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Theorem dalem57 39234
Description: Lemma for dath 39241. Axis of perspectivity point 𝐷 is on the auxiliary line 𝐡. (Contributed by NM, 9-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))))
dalem57.m ∧ = (meetβ€˜πΎ)
dalem57.o 𝑂 = (LPlanesβ€˜πΎ)
dalem57.y π‘Œ = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem57.z 𝑍 = ((𝑆 ∨ 𝑇) ∨ π‘ˆ)
dalem57.d 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))
dalem57.g 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
dalem57.h 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
dalem57.i 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ π‘ˆ))
dalem57.b1 𝐡 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ π‘Œ)
Assertion
Ref Expression
dalem57 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐷 ≀ 𝐡)

Proof of Theorem dalem57
StepHypRef Expression
1 dalem.ph . . . . . . 7 (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 . . . . . . 7 ≀ = (leβ€˜πΎ)
3 dalem.j . . . . . . 7 ∨ = (joinβ€˜πΎ)
4 dalem.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
5 dalem.ps . . . . . . 7 (πœ“ ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ Β¬ 𝑐 ≀ π‘Œ ∧ (𝑑 β‰  𝑐 ∧ Β¬ 𝑑 ≀ π‘Œ ∧ 𝐢 ≀ (𝑐 ∨ 𝑑))))
6 dalem57.m . . . . . . 7 ∧ = (meetβ€˜πΎ)
7 dalem57.o . . . . . . 7 𝑂 = (LPlanesβ€˜πΎ)
8 dalem57.y . . . . . . 7 π‘Œ = ((𝑃 ∨ 𝑄) ∨ 𝑅)
9 dalem57.z . . . . . . 7 𝑍 = ((𝑆 ∨ 𝑇) ∨ π‘ˆ)
10 dalem57.g . . . . . . 7 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
11 dalem57.h . . . . . . 7 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
12 dalem57.i . . . . . . 7 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ π‘ˆ))
13 dalem57.b1 . . . . . . 7 𝐡 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ π‘Œ)
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem55 39232 . . . . . 6 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐡))
151dalemkelat 39129 . . . . . . . 8 (πœ‘ β†’ 𝐾 ∈ Lat)
16153ad2ant1 1130 . . . . . . 7 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐾 ∈ Lat)
171dalemkehl 39128 . . . . . . . . 9 (πœ‘ β†’ 𝐾 ∈ HL)
18173ad2ant1 1130 . . . . . . . 8 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐾 ∈ HL)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 39201 . . . . . . . 8 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐺 ∈ 𝐴)
201, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 39206 . . . . . . . 8 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐻 ∈ 𝐴)
21 eqid 2728 . . . . . . . . 9 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2221, 3, 4hlatjcl 38871 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) β†’ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ))
2318, 19, 20, 22syl3anc 1368 . . . . . . 7 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ))
241, 3, 4dalempjqeb 39150 . . . . . . . 8 (πœ‘ β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
25243ad2ant1 1130 . . . . . . 7 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
2621, 2, 6latmle2 18464 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ)) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ (𝑃 ∨ 𝑄))
2716, 23, 25, 26syl3anc 1368 . . . . . 6 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ (𝑃 ∨ 𝑄))
2814, 27eqbrtrrd 5176 . . . . 5 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ (𝑃 ∨ 𝑄))
291, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem56 39233 . . . . . 6 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) = ((𝐺 ∨ 𝐻) ∧ 𝐡))
301, 3, 4dalemsjteb 39151 . . . . . . . 8 (πœ‘ β†’ (𝑆 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
31303ad2ant1 1130 . . . . . . 7 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (𝑆 ∨ 𝑇) ∈ (Baseβ€˜πΎ))
3221, 2, 6latmle2 18464 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ) ∧ (𝑆 ∨ 𝑇) ∈ (Baseβ€˜πΎ)) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) ≀ (𝑆 ∨ 𝑇))
3316, 23, 31, 32syl3anc 1368 . . . . . 6 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) ≀ (𝑆 ∨ 𝑇))
3429, 33eqbrtrrd 5176 . . . . 5 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ (𝑆 ∨ 𝑇))
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem54 39231 . . . . . . 7 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) ∈ 𝐴)
3621, 4atbase 38793 . . . . . . 7 (((𝐺 ∨ 𝐻) ∧ 𝐡) ∈ 𝐴 β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) ∈ (Baseβ€˜πΎ))
3735, 36syl 17 . . . . . 6 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) ∈ (Baseβ€˜πΎ))
3821, 2, 6latlem12 18465 . . . . . 6 ((𝐾 ∈ Lat ∧ (((𝐺 ∨ 𝐻) ∧ 𝐡) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ) ∧ (𝑆 ∨ 𝑇) ∈ (Baseβ€˜πΎ))) β†’ ((((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ (𝑃 ∨ 𝑄) ∧ ((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ (𝑆 ∨ 𝑇)) ↔ ((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))))
3916, 37, 25, 31, 38syl13anc 1369 . . . . 5 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ (𝑃 ∨ 𝑄) ∧ ((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ (𝑆 ∨ 𝑇)) ↔ ((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))))
4028, 34, 39mpbi2and 710 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)))
41 dalem57.d . . . 4 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))
4240, 41breqtrrdi 5194 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ 𝐷)
43 hlatl 38864 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
4418, 43syl 17 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐾 ∈ AtLat)
451, 2, 3, 4, 6, 7, 8, 9, 41dalemdea 39167 . . . . 5 (πœ‘ β†’ 𝐷 ∈ 𝐴)
46453ad2ant1 1130 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐷 ∈ 𝐴)
472, 4atcmp 38815 . . . 4 ((𝐾 ∈ AtLat ∧ ((𝐺 ∨ 𝐻) ∧ 𝐡) ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) β†’ (((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ 𝐷 ↔ ((𝐺 ∨ 𝐻) ∧ 𝐡) = 𝐷))
4844, 35, 46, 47syl3anc 1368 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ 𝐷 ↔ ((𝐺 ∨ 𝐻) ∧ 𝐡) = 𝐷))
4942, 48mpbid 231 . 2 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) = 𝐷)
50 eqid 2728 . . . . 5 (LLinesβ€˜πΎ) = (LLinesβ€˜πΎ)
511, 2, 3, 4, 5, 6, 50, 7, 8, 9, 10, 11, 12, 13dalem53 39230 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐡 ∈ (LLinesβ€˜πΎ))
5221, 50llnbase 39014 . . . 4 (𝐡 ∈ (LLinesβ€˜πΎ) β†’ 𝐡 ∈ (Baseβ€˜πΎ))
5351, 52syl 17 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐡 ∈ (Baseβ€˜πΎ))
5421, 2, 6latmle2 18464 . . 3 ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ) ∧ 𝐡 ∈ (Baseβ€˜πΎ)) β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ 𝐡)
5516, 23, 53, 54syl3anc 1368 . 2 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ 𝐡)
5649, 55eqbrtrrd 5176 1 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐷 ≀ 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  AtLatcal 38768  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:  dalem58  39235  dalem60  39237
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