Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem55 Structured version   Visualization version   GIF version
Theorem dalem55 38901
Description: Lemma for dath 38910. Lines 𝐺𝐻 and 𝑃𝑄 intersect at the auxiliary line 𝐡 (later shown to be an axis of perspectivity; see dalem60 38906). (Contributed by NM, 8-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))))
dalem54.m ∧ = (meetβ€˜πΎ)
dalem54.o 𝑂 = (LPlanesβ€˜πΎ)
dalem54.y π‘Œ = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem54.z 𝑍 = ((𝑆 ∨ 𝑇) ∨ π‘ˆ)
dalem54.g 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
dalem54.h 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
dalem54.i 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ π‘ˆ))
dalem54.b1 𝐡 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ π‘Œ)
Assertion
Ref Expression
dalem55 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐡))
Proof of Theorem dalem55
StepHypRef Expression
1 dalem.ph . . . . . 6 (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 38798 . . . . 5 (πœ‘ β†’ 𝐾 ∈ Lat)
323ad2ant1 1133 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐾 ∈ Lat)
41dalemkehl 38797 . . . . . 6 (πœ‘ β†’ 𝐾 ∈ HL)
543ad2ant1 1133 . . . . 5 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐾 ∈ HL)
6 dalem.l . . . . . 6 ≀ = (leβ€˜πΎ)
7 dalem.j . . . . . 6 ∨ = (joinβ€˜πΎ)
8 dalem.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
9 dalem.ps . . . . . 6 (πœ“ ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ Β¬ 𝑐 ≀ π‘Œ ∧ (𝑑 β‰  𝑐 ∧ Β¬ 𝑑 ≀ π‘Œ ∧ 𝐢 ≀ (𝑐 ∨ 𝑑))))
10 dalem54.m . . . . . 6 ∧ = (meetβ€˜πΎ)
11 dalem54.o . . . . . 6 𝑂 = (LPlanesβ€˜πΎ)
12 dalem54.y . . . . . 6 π‘Œ = ((𝑃 ∨ 𝑄) ∨ 𝑅)
13 dalem54.z . . . . . 6 𝑍 = ((𝑆 ∨ 𝑇) ∨ π‘ˆ)
14 dalem54.g . . . . . 6 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
151, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem23 38870 . . . . 5 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐺 ∈ 𝐴)
16 dalem54.h . . . . . 6 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
171, 6, 7, 8, 9, 10, 11, 12, 13, 16dalem29 38875 . . . . 5 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐻 ∈ 𝐴)
18 eqid 2732 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1918, 7, 8hlatjcl 38540 . . . . 5 ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) β†’ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ))
205, 15, 17, 19syl3anc 1371 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ))
211, 7, 8dalempjqeb 38819 . . . . 5 (πœ‘ β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
22213ad2ant1 1133 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
2318, 6, 10latmle1 18421 . . . 4 ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ)) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ (𝐺 ∨ 𝐻))
243, 20, 22, 23syl3anc 1371 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ (𝐺 ∨ 𝐻))
25 dalem54.i . . . . . . . 8 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ π‘ˆ))
261, 6, 7, 8, 9, 10, 11, 12, 13, 25dalem34 38880 . . . . . . 7 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐼 ∈ 𝐴)
2718, 8atbase 38462 . . . . . . 7 (𝐼 ∈ 𝐴 β†’ 𝐼 ∈ (Baseβ€˜πΎ))
2826, 27syl 17 . . . . . 6 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐼 ∈ (Baseβ€˜πΎ))
2918, 6, 7latlej1 18405 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ) ∧ 𝐼 ∈ (Baseβ€˜πΎ)) β†’ (𝐺 ∨ 𝐻) ≀ ((𝐺 ∨ 𝐻) ∨ 𝐼))
303, 20, 28, 29syl3anc 1371 . . . . 5 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (𝐺 ∨ 𝐻) ≀ ((𝐺 ∨ 𝐻) ∨ 𝐼))
311, 8dalemreb 38815 . . . . . . . 8 (πœ‘ β†’ 𝑅 ∈ (Baseβ€˜πΎ))
3218, 6, 7latlej1 18405 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ 𝑄) ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))
332, 21, 31, 32syl3anc 1371 . . . . . . 7 (πœ‘ β†’ (𝑃 ∨ 𝑄) ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))
3433, 12breqtrrdi 5190 . . . . . 6 (πœ‘ β†’ (𝑃 ∨ 𝑄) ≀ π‘Œ)
35343ad2ant1 1133 . . . . 5 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (𝑃 ∨ 𝑄) ≀ π‘Œ)
361, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25dalem42 38888 . . . . . . 7 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂)
3718, 11lplnbase 38708 . . . . . . 7 (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 β†’ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Baseβ€˜πΎ))
3836, 37syl 17 . . . . . 6 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Baseβ€˜πΎ))
391, 11dalemyeb 38823 . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ (Baseβ€˜πΎ))
40393ad2ant1 1133 . . . . . 6 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ π‘Œ ∈ (Baseβ€˜πΎ))
4118, 6, 10latmlem12 18428 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ) ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Baseβ€˜πΎ)) ∧ ((𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ) ∧ π‘Œ ∈ (Baseβ€˜πΎ))) β†’ (((𝐺 ∨ 𝐻) ≀ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ (𝑃 ∨ 𝑄) ≀ π‘Œ) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ π‘Œ)))
423, 20, 38, 22, 40, 41syl122anc 1379 . . . . 5 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (((𝐺 ∨ 𝐻) ≀ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ (𝑃 ∨ 𝑄) ≀ π‘Œ) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ π‘Œ)))
4330, 35, 42mp2and 697 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ π‘Œ))
44 dalem54.b1 . . . 4 𝐡 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ π‘Œ)
4543, 44breqtrrdi 5190 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ 𝐡)
4618, 10latmcl 18397 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ)) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Baseβ€˜πΎ))
473, 20, 22, 46syl3anc 1371 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Baseβ€˜πΎ))
48 eqid 2732 . . . . . 6 (LLinesβ€˜πΎ) = (LLinesβ€˜πΎ)
491, 6, 7, 8, 9, 10, 48, 11, 12, 13, 14, 16, 25, 44dalem53 38899 . . . . 5 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐡 ∈ (LLinesβ€˜πΎ))
5018, 48llnbase 38683 . . . . 5 (𝐡 ∈ (LLinesβ€˜πΎ) β†’ 𝐡 ∈ (Baseβ€˜πΎ))
5149, 50syl 17 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐡 ∈ (Baseβ€˜πΎ))
5218, 6, 10latlem12 18423 . . . 4 ((𝐾 ∈ Lat ∧ (((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Baseβ€˜πΎ) ∧ (𝐺 ∨ 𝐻) ∈ (Baseβ€˜πΎ) ∧ 𝐡 ∈ (Baseβ€˜πΎ))) β†’ ((((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ (𝐺 ∨ 𝐻) ∧ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ 𝐡) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ ((𝐺 ∨ 𝐻) ∧ 𝐡)))
533, 47, 20, 51, 52syl13anc 1372 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ (𝐺 ∨ 𝐻) ∧ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ 𝐡) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ ((𝐺 ∨ 𝐻) ∧ 𝐡)))
5424, 45, 53mpbi2and 710 . 2 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ ((𝐺 ∨ 𝐻) ∧ 𝐡))
55 hlatl 38533 . . . 4 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
565, 55syl 17 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐾 ∈ AtLat)
571, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25dalem52 38898 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
581, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25, 44dalem54 38900 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) ∈ 𝐴)
596, 8atcmp 38484 . . 3 ((𝐾 ∈ AtLat ∧ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ ((𝐺 ∨ 𝐻) ∧ 𝐡) ∈ 𝐴) β†’ (((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ ((𝐺 ∨ 𝐻) ∧ 𝐡) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐡)))
6056, 57, 58, 59syl3anc 1371 . 2 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ (((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≀ ((𝐺 ∨ 𝐻) ∧ 𝐡) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐡)))
6154, 60mpbid 231 1 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐡))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  lecple 17208  joincjn 18268  meetcmee 18269  Latclat 18388  Atomscatm 38436  AtLatcal 38437  HLchlt 38523  LLinesclln 38665  LPlanesclpl 38666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-proset 18252  df-poset 18270  df-plt 18287  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-p0 18382  df-lat 18389  df-clat 18456  df-oposet 38349  df-ol 38351  df-oml 38352  df-covers 38439  df-ats 38440  df-atl 38471  df-cvlat 38495  df-hlat 38524  df-llines 38672  df-lplanes 38673  df-lvols 38674
This theorem is referenced by:  dalem56  38902  dalem57  38903
  Copyright terms: Public domain W3C validator