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Theorem dalem60 36928
Description: Lemma for dath 36932. 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 36925 . . 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 36926 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐸 𝐵)
181dalemkelat 36820 . . . . 5 (𝜑𝐾 ∈ Lat)
19183ad2ant1 1130 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
201, 2, 3, 4, 6, 7, 8, 9, 10dalemdea 36858 . . . . . 6 (𝜑𝐷𝐴)
21 eqid 2824 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2221, 4atbase 36485 . . . . . 6 (𝐷𝐴𝐷 ∈ (Base‘𝐾))
2320, 22syl 17 . . . . 5 (𝜑𝐷 ∈ (Base‘𝐾))
24233ad2ant1 1130 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐷 ∈ (Base‘𝐾))
251, 2, 3, 4, 6, 7, 8, 9, 16dalemeea 36859 . . . . . 6 (𝜑𝐸𝐴)
2621, 4atbase 36485 . . . . . 6 (𝐸𝐴𝐸 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . 5 (𝜑𝐸 ∈ (Base‘𝐾))
28273ad2ant1 1130 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐸 ∈ (Base‘𝐾))
29 eqid 2824 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
301, 2, 3, 4, 5, 6, 29, 7, 8, 9, 11, 12, 13, 14dalem53 36921 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
3121, 29llnbase 36705 . . . . 5 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
3230, 31syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
3321, 2, 3latjle12 17661 . . . 4 ((𝐾 ∈ Lat ∧ (𝐷 ∈ (Base‘𝐾) ∧ 𝐸 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾))) → ((𝐷 𝐵𝐸 𝐵) ↔ (𝐷 𝐸) 𝐵))
3419, 24, 28, 32, 33syl13anc 1369 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐷 𝐵𝐸 𝐵) ↔ (𝐷 𝐸) 𝐵))
3515, 17, 34mpbi2and 711 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) 𝐵)
361dalemkehl 36819 . . . 4 (𝜑𝐾 ∈ HL)
37363ad2ant1 1130 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
381, 2, 3, 4, 6, 7, 8, 9, 10, 16dalemdnee 36862 . . . . 5 (𝜑𝐷𝐸)
393, 4, 29llni2 36708 . . . . 5 (((𝐾 ∈ HL ∧ 𝐷𝐴𝐸𝐴) ∧ 𝐷𝐸) → (𝐷 𝐸) ∈ (LLines‘𝐾))
4036, 20, 25, 38, 39syl31anc 1370 . . . 4 (𝜑 → (𝐷 𝐸) ∈ (LLines‘𝐾))
41403ad2ant1 1130 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) ∈ (LLines‘𝐾))
422, 29llncmp 36718 . . 3 ((𝐾 ∈ HL ∧ (𝐷 𝐸) ∈ (LLines‘𝐾) ∧ 𝐵 ∈ (LLines‘𝐾)) → ((𝐷 𝐸) 𝐵 ↔ (𝐷 𝐸) = 𝐵))
4337, 41, 30, 42syl3anc 1368 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐷 𝐸) 𝐵 ↔ (𝐷 𝐸) = 𝐵))
4435, 43mpbid 235 1 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3013   class class class wbr 5047  cfv 6336  (class class class)co 7138  Basecbs 16472  lecple 16561  joincjn 17543  meetcmee 17544  Latclat 17644  Atomscatm 36459  HLchlt 36546  LLinesclln 36687  LPlanesclpl 36688
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-proset 17527  df-poset 17545  df-plt 17557  df-lub 17573  df-glb 17574  df-join 17575  df-meet 17576  df-p0 17638  df-lat 17645  df-clat 17707  df-oposet 36372  df-ol 36374  df-oml 36375  df-covers 36462  df-ats 36463  df-atl 36494  df-cvlat 36518  df-hlat 36547  df-llines 36694  df-lplanes 36695  df-lvols 36696
This theorem is referenced by:  dalem61  36929
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