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Theorem dalem60 35540
Description: Lemma for dath 35544. 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 35537 . . 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 35538 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐸 𝐵)
181dalemkelat 35432 . . . . 5 (𝜑𝐾 ∈ Lat)
19183ad2ant1 1127 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
201, 2, 3, 4, 6, 7, 8, 9, 10dalemdea 35470 . . . . . 6 (𝜑𝐷𝐴)
21 eqid 2771 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2221, 4atbase 35098 . . . . . 6 (𝐷𝐴𝐷 ∈ (Base‘𝐾))
2320, 22syl 17 . . . . 5 (𝜑𝐷 ∈ (Base‘𝐾))
24233ad2ant1 1127 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐷 ∈ (Base‘𝐾))
251, 2, 3, 4, 6, 7, 8, 9, 16dalemeea 35471 . . . . . 6 (𝜑𝐸𝐴)
2621, 4atbase 35098 . . . . . 6 (𝐸𝐴𝐸 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . 5 (𝜑𝐸 ∈ (Base‘𝐾))
28273ad2ant1 1127 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐸 ∈ (Base‘𝐾))
29 eqid 2771 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
301, 2, 3, 4, 5, 6, 29, 7, 8, 9, 11, 12, 13, 14dalem53 35533 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
3121, 29llnbase 35317 . . . . 5 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
3230, 31syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
3321, 2, 3latjle12 17270 . . . 4 ((𝐾 ∈ Lat ∧ (𝐷 ∈ (Base‘𝐾) ∧ 𝐸 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾))) → ((𝐷 𝐵𝐸 𝐵) ↔ (𝐷 𝐸) 𝐵))
3419, 24, 28, 32, 33syl13anc 1478 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐷 𝐵𝐸 𝐵) ↔ (𝐷 𝐸) 𝐵))
3515, 17, 34mpbi2and 691 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) 𝐵)
361dalemkehl 35431 . . . 4 (𝜑𝐾 ∈ HL)
37363ad2ant1 1127 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
381, 2, 3, 4, 6, 7, 8, 9, 10, 16dalemdnee 35474 . . . . 5 (𝜑𝐷𝐸)
393, 4, 29llni2 35320 . . . . 5 (((𝐾 ∈ HL ∧ 𝐷𝐴𝐸𝐴) ∧ 𝐷𝐸) → (𝐷 𝐸) ∈ (LLines‘𝐾))
4036, 20, 25, 38, 39syl31anc 1479 . . . 4 (𝜑 → (𝐷 𝐸) ∈ (LLines‘𝐾))
41403ad2ant1 1127 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) ∈ (LLines‘𝐾))
422, 29llncmp 35330 . . 3 ((𝐾 ∈ HL ∧ (𝐷 𝐸) ∈ (LLines‘𝐾) ∧ 𝐵 ∈ (LLines‘𝐾)) → ((𝐷 𝐸) 𝐵 ↔ (𝐷 𝐸) = 𝐵))
4337, 41, 30, 42syl3anc 1476 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐷 𝐸) 𝐵 ↔ (𝐷 𝐸) = 𝐵))
4435, 43mpbid 222 1 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Latclat 17253  Atomscatm 35072  HLchlt 35159  LLinesclln 35299  LPlanesclpl 35300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306  df-lplanes 35307  df-lvols 35308
This theorem is referenced by:  dalem61  35541
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