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Theorem dalem57 34033
 Description: Lemma for dath 34040. 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 34031 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵))
151dalemkelat 33928 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
16153ad2ant1 1075 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
171dalemkehl 33927 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
18173ad2ant1 1075 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 34000 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
201, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 34005 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
21 eqid 2610 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2221, 3, 4hlatjcl 33671 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
2318, 19, 20, 22syl3anc 1318 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
241, 3, 4dalempjqeb 33949 . . . . . . . 8 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
25243ad2ant1 1075 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
2621, 2, 6latmle2 16900 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) (𝑃 𝑄))
2716, 23, 25, 26syl3anc 1318 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (𝑃 𝑄))
2814, 27eqbrtrrd 4607 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) (𝑃 𝑄))
291, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem56 34032 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) = ((𝐺 𝐻) 𝐵))
301, 3, 4dalemsjteb 33950 . . . . . . . 8 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
31303ad2ant1 1075 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝑆 𝑇) ∈ (Base‘𝐾))
3221, 2, 6latmle2 16900 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑆 𝑇)) (𝑆 𝑇))
3316, 23, 31, 32syl3anc 1318 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) (𝑆 𝑇))
3429, 33eqbrtrrd 4607 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) (𝑆 𝑇))
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem54 34030 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)
3621, 4atbase 33594 . . . . . . 7 (((𝐺 𝐻) 𝐵) ∈ 𝐴 → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
3735, 36syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
3821, 2, 6latlem12 16901 . . . . . 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 1320 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐺 𝐻) 𝐵) (𝑃 𝑄) ∧ ((𝐺 𝐻) 𝐵) (𝑆 𝑇)) ↔ ((𝐺 𝐻) 𝐵) ((𝑃 𝑄) (𝑆 𝑇))))
4028, 34, 39mpbi2and 958 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ((𝑃 𝑄) (𝑆 𝑇)))
41 dalem57.d . . . 4 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
4240, 41syl6breqr 4625 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) 𝐷)
43 hlatl 33665 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
4418, 43syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
451, 2, 3, 4, 6, 7, 8, 9, 41dalemdea 33966 . . . . 5 (𝜑𝐷𝐴)
46453ad2ant1 1075 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐷𝐴)
472, 4atcmp 33616 . . . 4 ((𝐾 ∈ AtLat ∧ ((𝐺 𝐻) 𝐵) ∈ 𝐴𝐷𝐴) → (((𝐺 𝐻) 𝐵) 𝐷 ↔ ((𝐺 𝐻) 𝐵) = 𝐷))
4844, 35, 46, 47syl3anc 1318 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐵) 𝐷 ↔ ((𝐺 𝐻) 𝐵) = 𝐷))
4942, 48mpbid 221 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) = 𝐷)
50 eqid 2610 . . . . 5 (LLines‘𝐾) = (LLines‘𝐾)
511, 2, 3, 4, 5, 6, 50, 7, 8, 9, 10, 11, 12, 13dalem53 34029 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
5221, 50llnbase 33813 . . . 4 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
5351, 52syl 17 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
5421, 2, 6latmle2 16900 . . 3 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐵) 𝐵)
5516, 23, 53, 54syl3anc 1318 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) 𝐵)
5649, 55eqbrtrrd 4607 1 ((𝜑𝑌 = 𝑍𝜓) → 𝐷 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Latclat 16868  Atomscatm 33568  AtLatcal 33569  HLchlt 33655  LLinesclln 33795  LPlanesclpl 33796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804 This theorem is referenced by:  dalem58  34034  dalem60  34036
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