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Theorem dalem60 34036
 Description: Lemma for dath 34040. 𝐵 is an axis of perspectivity (almost). (Contributed by NM, 11-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))))
dalem60.m = (meet‘𝐾)
dalem60.o 𝑂 = (LPlanes‘𝐾)
dalem60.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem60.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem60.d 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
dalem60.e 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
dalem60.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
dalem60.h 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
dalem60.i 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
dalem60.b1 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
Assertion
Ref Expression
dalem60 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) = 𝐵)

Proof of Theorem dalem60
StepHypRef Expression
1 dalem.ph . . . 4 (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 . . . 4 = (le‘𝐾)
3 dalem.j . . . 4 = (join‘𝐾)
4 dalem.a . . . 4 𝐴 = (Atoms‘𝐾)
5 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
6 dalem60.m . . . 4 = (meet‘𝐾)
7 dalem60.o . . . 4 𝑂 = (LPlanes‘𝐾)
8 dalem60.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
9 dalem60.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
10 dalem60.d . . . 4 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
11 dalem60.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
12 dalem60.h . . . 4 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
13 dalem60.i . . . 4 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
14 dalem60.b1 . . . 4 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem57 34033 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐷 𝐵)
16 dalem60.e . . . 4 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
171, 2, 3, 4, 5, 6, 7, 8, 9, 16, 11, 12, 13, 14dalem58 34034 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐸 𝐵)
181dalemkelat 33928 . . . . 5 (𝜑𝐾 ∈ Lat)
19183ad2ant1 1075 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
201, 2, 3, 4, 6, 7, 8, 9, 10dalemdea 33966 . . . . . 6 (𝜑𝐷𝐴)
21 eqid 2610 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2221, 4atbase 33594 . . . . . 6 (𝐷𝐴𝐷 ∈ (Base‘𝐾))
2320, 22syl 17 . . . . 5 (𝜑𝐷 ∈ (Base‘𝐾))
24233ad2ant1 1075 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐷 ∈ (Base‘𝐾))
251, 2, 3, 4, 6, 7, 8, 9, 16dalemeea 33967 . . . . . 6 (𝜑𝐸𝐴)
2621, 4atbase 33594 . . . . . 6 (𝐸𝐴𝐸 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . 5 (𝜑𝐸 ∈ (Base‘𝐾))
28273ad2ant1 1075 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐸 ∈ (Base‘𝐾))
29 eqid 2610 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
301, 2, 3, 4, 5, 6, 29, 7, 8, 9, 11, 12, 13, 14dalem53 34029 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
3121, 29llnbase 33813 . . . . 5 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
3230, 31syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
3321, 2, 3latjle12 16885 . . . 4 ((𝐾 ∈ Lat ∧ (𝐷 ∈ (Base‘𝐾) ∧ 𝐸 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾))) → ((𝐷 𝐵𝐸 𝐵) ↔ (𝐷 𝐸) 𝐵))
3419, 24, 28, 32, 33syl13anc 1320 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐷 𝐵𝐸 𝐵) ↔ (𝐷 𝐸) 𝐵))
3515, 17, 34mpbi2and 958 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) 𝐵)
361dalemkehl 33927 . . . 4 (𝜑𝐾 ∈ HL)
37363ad2ant1 1075 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
381, 2, 3, 4, 6, 7, 8, 9, 10, 16dalemdnee 33970 . . . . 5 (𝜑𝐷𝐸)
393, 4, 29llni2 33816 . . . . 5 (((𝐾 ∈ HL ∧ 𝐷𝐴𝐸𝐴) ∧ 𝐷𝐸) → (𝐷 𝐸) ∈ (LLines‘𝐾))
4036, 20, 25, 38, 39syl31anc 1321 . . . 4 (𝜑 → (𝐷 𝐸) ∈ (LLines‘𝐾))
41403ad2ant1 1075 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) ∈ (LLines‘𝐾))
422, 29llncmp 33826 . . 3 ((𝐾 ∈ HL ∧ (𝐷 𝐸) ∈ (LLines‘𝐾) ∧ 𝐵 ∈ (LLines‘𝐾)) → ((𝐷 𝐸) 𝐵 ↔ (𝐷 𝐸) = 𝐵))
4337, 41, 30, 42syl3anc 1318 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐷 𝐸) 𝐵 ↔ (𝐷 𝐸) = 𝐵))
4435, 43mpbid 221 1 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) = 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  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:  dalem61  34037
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