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 39090
Description: Lemma for dath 39097. Axis of perspectivity point 𝐷 is on the auxiliary line 𝐵. (Contributed by NM, 9-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))))
dalem57.m = (meet‘𝐾)
dalem57.o 𝑂 = (LPlanes‘𝐾)
dalem57.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem57.z 𝑍 = ((𝑆 𝑇) 𝑈)
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 𝑈))
dalem57.b1 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
Assertion
Ref Expression
dalem57 ((𝜑𝑌 = 𝑍𝜓) → 𝐷 𝐵)

Proof of Theorem dalem57
StepHypRef Expression
1 dalem.ph . . . . . . 7 (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)))))
2 dalem.l . . . . . . 7 = (le‘𝐾)
3 dalem.j . . . . . . 7 = (join‘𝐾)
4 dalem.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
5 dalem.ps . . . . . . 7 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
6 dalem57.m . . . . . . 7 = (meet‘𝐾)
7 dalem57.o . . . . . . 7 𝑂 = (LPlanes‘𝐾)
8 dalem57.y . . . . . . 7 𝑌 = ((𝑃 𝑄) 𝑅)
9 dalem57.z . . . . . . 7 𝑍 = ((𝑆 𝑇) 𝑈)
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 𝑈))
13 dalem57.b1 . . . . . . 7 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem55 39088 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵))
151dalemkelat 38985 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
16153ad2ant1 1130 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
171dalemkehl 38984 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
18173ad2ant1 1130 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 39057 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
201, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 39062 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
21 eqid 2724 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2221, 3, 4hlatjcl 38727 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
2318, 19, 20, 22syl3anc 1368 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
241, 3, 4dalempjqeb 39006 . . . . . . . 8 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
25243ad2ant1 1130 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
2621, 2, 6latmle2 18420 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) (𝑃 𝑄))
2716, 23, 25, 26syl3anc 1368 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (𝑃 𝑄))
2814, 27eqbrtrrd 5162 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) (𝑃 𝑄))
291, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem56 39089 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) = ((𝐺 𝐻) 𝐵))
301, 3, 4dalemsjteb 39007 . . . . . . . 8 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
31303ad2ant1 1130 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝑆 𝑇) ∈ (Base‘𝐾))
3221, 2, 6latmle2 18420 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑆 𝑇)) (𝑆 𝑇))
3316, 23, 31, 32syl3anc 1368 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) (𝑆 𝑇))
3429, 33eqbrtrrd 5162 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) (𝑆 𝑇))
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem54 39087 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)
3621, 4atbase 38649 . . . . . . 7 (((𝐺 𝐻) 𝐵) ∈ 𝐴 → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
3735, 36syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
3821, 2, 6latlem12 18421 . . . . . 6 ((𝐾 ∈ Lat ∧ (((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾))) → ((((𝐺 𝐻) 𝐵) (𝑃 𝑄) ∧ ((𝐺 𝐻) 𝐵) (𝑆 𝑇)) ↔ ((𝐺 𝐻) 𝐵) ((𝑃 𝑄) (𝑆 𝑇))))
3916, 37, 25, 31, 38syl13anc 1369 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐺 𝐻) 𝐵) (𝑃 𝑄) ∧ ((𝐺 𝐻) 𝐵) (𝑆 𝑇)) ↔ ((𝐺 𝐻) 𝐵) ((𝑃 𝑄) (𝑆 𝑇))))
4028, 34, 39mpbi2and 709 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ((𝑃 𝑄) (𝑆 𝑇)))
41 dalem57.d . . . 4 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
4240, 41breqtrrdi 5180 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) 𝐷)
43 hlatl 38720 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
4418, 43syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
451, 2, 3, 4, 6, 7, 8, 9, 41dalemdea 39023 . . . . 5 (𝜑𝐷𝐴)
46453ad2ant1 1130 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐷𝐴)
472, 4atcmp 38671 . . . 4 ((𝐾 ∈ AtLat ∧ ((𝐺 𝐻) 𝐵) ∈ 𝐴𝐷𝐴) → (((𝐺 𝐻) 𝐵) 𝐷 ↔ ((𝐺 𝐻) 𝐵) = 𝐷))
4844, 35, 46, 47syl3anc 1368 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐵) 𝐷 ↔ ((𝐺 𝐻) 𝐵) = 𝐷))
4942, 48mpbid 231 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) = 𝐷)
50 eqid 2724 . . . . 5 (LLines‘𝐾) = (LLines‘𝐾)
511, 2, 3, 4, 5, 6, 50, 7, 8, 9, 10, 11, 12, 13dalem53 39086 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
5221, 50llnbase 38870 . . . 4 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
5351, 52syl 17 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
5421, 2, 6latmle2 18420 . . 3 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐵) 𝐵)
5516, 23, 53, 54syl3anc 1368 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) 𝐵)
5649, 55eqbrtrrd 5162 1 ((𝜑𝑌 = 𝑍𝜓) → 𝐷 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wne 2932   class class class wbr 5138  cfv 6533  (class class class)co 7401  Basecbs 17143  lecple 17203  joincjn 18266  meetcmee 18267  Latclat 18386  Atomscatm 38623  AtLatcal 38624  HLchlt 38710  LLinesclln 38852  LPlanesclpl 38853
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 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  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 38536  df-ol 38538  df-oml 38539  df-covers 38626  df-ats 38627  df-atl 38658  df-cvlat 38682  df-hlat 38711  df-llines 38859  df-lplanes 38860  df-lvols 38861
This theorem is referenced by:  dalem58  39091  dalem60  39093
  Copyright terms: Public domain W3C validator