Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem57 Structured version   Visualization version   GIF version

Theorem dalem57 39112
Description: Lemma for dath 39119. 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 39110 . . . . . 6 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐡))
151dalemkelat 39007 . . . . . . . 8 (πœ‘ β†’ 𝐾 ∈ Lat)
16153ad2ant1 1130 . . . . . . 7 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐾 ∈ Lat)
171dalemkehl 39006 . . . . . . . . 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 39079 . . . . . . . 8 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐺 ∈ 𝐴)
201, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 39084 . . . . . . . 8 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐻 ∈ 𝐴)
21 eqid 2726 . . . . . . . . 9 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2221, 3, 4hlatjcl 38749 . . . . . . . 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 39028 . . . . . . . 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 18427 . . . . . . 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 5165 . . . . 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 39111 . . . . . 6 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) = ((𝐺 ∨ 𝐻) ∧ 𝐡))
301, 3, 4dalemsjteb 39029 . . . . . . . 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 18427 . . . . . . 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 5165 . . . . 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 39109 . . . . . . 7 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) ∈ 𝐴)
3621, 4atbase 38671 . . . . . . 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 18428 . . . . . 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 709 . . . 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 5183 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) ≀ 𝐷)
43 hlatl 38742 . . . . 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 39045 . . . . 5 (πœ‘ β†’ 𝐷 ∈ 𝐴)
46453ad2ant1 1130 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐷 ∈ 𝐴)
472, 4atcmp 38693 . . . 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 2726 . . . . 5 (LLinesβ€˜πΎ) = (LLinesβ€˜πΎ)
511, 2, 3, 4, 5, 6, 50, 7, 8, 9, 10, 11, 12, 13dalem53 39108 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐡 ∈ (LLinesβ€˜πΎ))
5221, 50llnbase 38892 . . . 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 18427 . . 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 5165 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  Basecbs 17150  lecple 17210  joincjn 18273  meetcmee 18274  Latclat 18393  Atomscatm 38645  AtLatcal 38646  HLchlt 38732  LLinesclln 38874  LPlanesclpl 38875
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-proset 18257  df-poset 18275  df-plt 18292  df-lub 18308  df-glb 18309  df-join 18310  df-meet 18311  df-p0 18387  df-lat 18394  df-clat 18461  df-oposet 38558  df-ol 38560  df-oml 38561  df-covers 38648  df-ats 38649  df-atl 38680  df-cvlat 38704  df-hlat 38733  df-llines 38881  df-lplanes 38882  df-lvols 38883
This theorem is referenced by:  dalem58  39113  dalem60  39115
  Copyright terms: Public domain W3C validator