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 39710
Description: Lemma for dath 39719. Lines 𝐺𝐻 and 𝑃𝑄 intersect at the auxiliary line 𝐵 (later shown to be an axis of perspectivity; see dalem60 39715). (Contributed by NM, 8-Aug-2012.)
Hypotheses
Ref Expression
dalem.ph (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
dalem.l = (le‘𝐾)
dalem.j = (join‘𝐾)
dalem.a 𝐴 = (Atoms‘𝐾)
dalem.ps (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
dalem54.m = (meet‘𝐾)
dalem54.o 𝑂 = (LPlanes‘𝐾)
dalem54.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem54.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem54.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
dalem54.h 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
dalem54.i 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
dalem54.b1 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
Assertion
Ref Expression
dalem55 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵))

Proof of Theorem dalem55
StepHypRef Expression
1 dalem.ph . . . . . 6 (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
21dalemkelat 39607 . . . . 5 (𝜑𝐾 ∈ Lat)
323ad2ant1 1132 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
41dalemkehl 39606 . . . . . 6 (𝜑𝐾 ∈ HL)
543ad2ant1 1132 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
6 dalem.l . . . . . 6 = (le‘𝐾)
7 dalem.j . . . . . 6 = (join‘𝐾)
8 dalem.a . . . . . 6 𝐴 = (Atoms‘𝐾)
9 dalem.ps . . . . . 6 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
10 dalem54.m . . . . . 6 = (meet‘𝐾)
11 dalem54.o . . . . . 6 𝑂 = (LPlanes‘𝐾)
12 dalem54.y . . . . . 6 𝑌 = ((𝑃 𝑄) 𝑅)
13 dalem54.z . . . . . 6 𝑍 = ((𝑆 𝑇) 𝑈)
14 dalem54.g . . . . . 6 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
151, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem23 39679 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
16 dalem54.h . . . . . 6 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
171, 6, 7, 8, 9, 10, 11, 12, 13, 16dalem29 39684 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
18 eqid 2735 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
1918, 7, 8hlatjcl 39349 . . . . 5 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
205, 15, 17, 19syl3anc 1370 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
211, 7, 8dalempjqeb 39628 . . . . 5 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
22213ad2ant1 1132 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
2318, 6, 10latmle1 18522 . . . 4 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻))
243, 20, 22, 23syl3anc 1370 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻))
25 dalem54.i . . . . . . . 8 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
261, 6, 7, 8, 9, 10, 11, 12, 13, 25dalem34 39689 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2718, 8atbase 39271 . . . . . . 7 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
2826, 27syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
2918, 6, 7latlej1 18506 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → (𝐺 𝐻) ((𝐺 𝐻) 𝐼))
303, 20, 28, 29syl3anc 1370 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ((𝐺 𝐻) 𝐼))
311, 8dalemreb 39624 . . . . . . . 8 (𝜑𝑅 ∈ (Base‘𝐾))
3218, 6, 7latlej1 18506 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑅))
332, 21, 31, 32syl3anc 1370 . . . . . . 7 (𝜑 → (𝑃 𝑄) ((𝑃 𝑄) 𝑅))
3433, 12breqtrrdi 5190 . . . . . 6 (𝜑 → (𝑃 𝑄) 𝑌)
35343ad2ant1 1132 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) 𝑌)
361, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25dalem42 39697 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
3718, 11lplnbase 39517 . . . . . . 7 (((𝐺 𝐻) 𝐼) ∈ 𝑂 → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
3836, 37syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
391, 11dalemyeb 39632 . . . . . . 7 (𝜑𝑌 ∈ (Base‘𝐾))
40393ad2ant1 1132 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
4118, 6, 10latmlem12 18529 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) ∈ (Base‘𝐾) ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾)) ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → (((𝐺 𝐻) ((𝐺 𝐻) 𝐼) ∧ (𝑃 𝑄) 𝑌) → ((𝐺 𝐻) (𝑃 𝑄)) (((𝐺 𝐻) 𝐼) 𝑌)))
423, 20, 38, 22, 40, 41syl122anc 1378 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) ((𝐺 𝐻) 𝐼) ∧ (𝑃 𝑄) 𝑌) → ((𝐺 𝐻) (𝑃 𝑄)) (((𝐺 𝐻) 𝐼) 𝑌)))
4330, 35, 42mp2and 699 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (((𝐺 𝐻) 𝐼) 𝑌))
44 dalem54.b1 . . . 4 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
4543, 44breqtrrdi 5190 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) 𝐵)
4618, 10latmcl 18498 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
473, 20, 22, 46syl3anc 1370 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
48 eqid 2735 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
491, 6, 7, 8, 9, 10, 48, 11, 12, 13, 14, 16, 25, 44dalem53 39708 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
5018, 48llnbase 39492 . . . . 5 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
5149, 50syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
5218, 6, 10latlem12 18524 . . . 4 ((𝐾 ∈ Lat ∧ (((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾) ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾))) → ((((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻) ∧ ((𝐺 𝐻) (𝑃 𝑄)) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵)))
533, 47, 20, 51, 52syl13anc 1371 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻) ∧ ((𝐺 𝐻) (𝑃 𝑄)) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵)))
5424, 45, 53mpbi2and 712 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵))
55 hlatl 39342 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
565, 55syl 17 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
571, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25dalem52 39707 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)
581, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25, 44dalem54 39709 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)
596, 8atcmp 39293 . . 3 ((𝐾 ∈ AtLat ∧ ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴 ∧ ((𝐺 𝐻) 𝐵) ∈ 𝐴) → (((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵)))
6056, 57, 58, 59syl3anc 1370 . 2 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵)))
6154, 60mpbid 232 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Latclat 18489  Atomscatm 39245  AtLatcal 39246  HLchlt 39332  LLinesclln 39474  LPlanesclpl 39475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482  df-lvols 39483
This theorem is referenced by:  dalem56  39711  dalem57  39712
  Copyright terms: Public domain W3C validator