Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem45 Structured version   Visualization version   GIF version

Theorem dalem45 39099
Description: Lemma for dath 39118. Dummy center of perspectivity 𝑐 is not on the line 𝐺𝐻. (Contributed by NM, 16-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))))
dalem44.m ∧ = (meetβ€˜πΎ)
dalem44.o 𝑂 = (LPlanesβ€˜πΎ)
dalem44.y π‘Œ = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem44.z 𝑍 = ((𝑆 ∨ 𝑇) ∨ π‘ˆ)
dalem44.g 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
dalem44.h 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
dalem44.i 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ π‘ˆ))
Assertion
Ref Expression
dalem45 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ Β¬ 𝑐 ≀ (𝐺 ∨ 𝐻))

Proof of Theorem dalem45
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)))))
21dalemkelat 39006 . . 3 (πœ‘ β†’ 𝐾 ∈ Lat)
323ad2ant1 1130 . 2 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐾 ∈ Lat)
4 dalem.ps . . . 4 (πœ“ ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ Β¬ 𝑐 ≀ π‘Œ ∧ (𝑑 β‰  𝑐 ∧ Β¬ 𝑑 ≀ π‘Œ ∧ 𝐢 ≀ (𝑐 ∨ 𝑑))))
5 dalem.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
64, 5dalemcceb 39071 . . 3 (πœ“ β†’ 𝑐 ∈ (Baseβ€˜πΎ))
763ad2ant3 1132 . 2 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝑐 ∈ (Baseβ€˜πΎ))
81dalemkehl 39005 . . . 4 (πœ‘ β†’ 𝐾 ∈ HL)
983ad2ant1 1130 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐾 ∈ HL)
10 dalem.l . . . 4 ≀ = (leβ€˜πΎ)
11 dalem.j . . . 4 ∨ = (joinβ€˜πΎ)
12 dalem44.m . . . 4 ∧ = (meetβ€˜πΎ)
13 dalem44.o . . . 4 𝑂 = (LPlanesβ€˜πΎ)
14 dalem44.y . . . 4 π‘Œ = ((𝑃 ∨ 𝑄) ∨ 𝑅)
15 dalem44.z . . . 4 𝑍 = ((𝑆 ∨ 𝑇) ∨ π‘ˆ)
16 dalem44.g . . . 4 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
171, 10, 11, 5, 4, 12, 13, 14, 15, 16dalem23 39078 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐺 ∈ 𝐴)
18 dalem44.h . . . 4 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
191, 10, 11, 5, 4, 12, 13, 14, 15, 18dalem29 39083 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐻 ∈ 𝐴)
20 eqid 2726 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2120, 11, 5hlatjcl 38748 . . 3 ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) β†’ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ))
229, 17, 19, 21syl3anc 1368 . 2 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ))
23 dalem44.i . . . 4 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ π‘ˆ))
241, 10, 11, 5, 4, 12, 13, 14, 15, 23dalem34 39088 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐼 ∈ 𝐴)
2520, 5atbase 38670 . . 3 (𝐼 ∈ 𝐴 β†’ 𝐼 ∈ (Baseβ€˜πΎ))
2624, 25syl 17 . 2 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐼 ∈ (Baseβ€˜πΎ))
271, 10, 11, 5, 4, 12, 13, 14, 15, 16, 18, 23dalem44 39098 . 2 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ Β¬ 𝑐 ≀ ((𝐺 ∨ 𝐻) ∨ 𝐼))
2820, 10, 11latnlej2l 18423 . 2 ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Baseβ€˜πΎ) ∧ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ) ∧ 𝐼 ∈ (Baseβ€˜πΎ)) ∧ Β¬ 𝑐 ≀ ((𝐺 ∨ 𝐻) ∨ 𝐼)) β†’ Β¬ 𝑐 ≀ (𝐺 ∨ 𝐻))
293, 7, 22, 26, 27, 28syl131anc 1380 1 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ Β¬ 𝑐 ≀ (𝐺 ∨ 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 17151  lecple 17211  joincjn 18274  meetcmee 18275  Latclat 18394  Atomscatm 38644  HLchlt 38731  LPlanesclpl 38874
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-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 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-lat 18395  df-clat 18462  df-oposet 38557  df-ol 38559  df-oml 38560  df-covers 38647  df-ats 38648  df-atl 38679  df-cvlat 38703  df-hlat 38732  df-llines 38880  df-lplanes 38881  df-lvols 38882
This theorem is referenced by:  dalem46  39100  dalem51  39105  dalem52  39106
  Copyright terms: Public domain W3C validator