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Theorem dalem55 38684
Description: Lemma for dath 38693. Lines 𝐺𝐻 and 𝑃𝑄 intersect at the auxiliary line 𝐡 (later shown to be an axis of perspectivity; see dalem60 38689). (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 38581 . . . . 5 (πœ‘ β†’ 𝐾 ∈ Lat)
323ad2ant1 1133 . . . 4 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐾 ∈ Lat)
41dalemkehl 38580 . . . . . 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 38653 . . . . 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 38658 . . . . 5 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐻 ∈ 𝐴)
18 eqid 2732 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1918, 7, 8hlatjcl 38323 . . . . 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 38602 . . . . 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 18419 . . . 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 38663 . . . . . . 7 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐼 ∈ 𝐴)
2718, 8atbase 38245 . . . . . . 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 18403 . . . . . 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 38598 . . . . . . . 8 (πœ‘ β†’ 𝑅 ∈ (Baseβ€˜πΎ))
3218, 6, 7latlej1 18403 . . . . . . . 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 38671 . . . . . . 7 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂)
3718, 11lplnbase 38491 . . . . . . 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 38606 . . . . . . 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 18426 . . . . . 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 18395 . . . . 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 38682 . . . . 5 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ 𝐡 ∈ (LLinesβ€˜πΎ))
5018, 48llnbase 38466 . . . . 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 18421 . . . 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 38316 . . . 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 38681 . . 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 38683 . . 3 ((πœ‘ ∧ π‘Œ = 𝑍 ∧ πœ“) β†’ ((𝐺 ∨ 𝐻) ∧ 𝐡) ∈ 𝐴)
596, 8atcmp 38267 . . 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 17146  lecple 17206  joincjn 18266  meetcmee 18267  Latclat 18386  Atomscatm 38219  AtLatcal 38220  HLchlt 38306  LLinesclln 38448  LPlanesclpl 38449
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 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-lat 18387  df-clat 18454  df-oposet 38132  df-ol 38134  df-oml 38135  df-covers 38222  df-ats 38223  df-atl 38254  df-cvlat 38278  df-hlat 38307  df-llines 38455  df-lplanes 38456  df-lvols 38457
This theorem is referenced by:  dalem56  38685  dalem57  38686
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