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Theorem dalem57 37398
Description: Lemma for dath 37405. 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 37396 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵))
151dalemkelat 37293 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
16153ad2ant1 1134 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
171dalemkehl 37292 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
18173ad2ant1 1134 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 37365 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
201, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 37370 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
21 eqid 2739 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2221, 3, 4hlatjcl 37036 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
2318, 19, 20, 22syl3anc 1372 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
241, 3, 4dalempjqeb 37314 . . . . . . . 8 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
25243ad2ant1 1134 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
2621, 2, 6latmle2 17815 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) (𝑃 𝑄))
2716, 23, 25, 26syl3anc 1372 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (𝑃 𝑄))
2814, 27eqbrtrrd 5064 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) (𝑃 𝑄))
291, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem56 37397 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) = ((𝐺 𝐻) 𝐵))
301, 3, 4dalemsjteb 37315 . . . . . . . 8 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
31303ad2ant1 1134 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝑆 𝑇) ∈ (Base‘𝐾))
3221, 2, 6latmle2 17815 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑆 𝑇)) (𝑆 𝑇))
3316, 23, 31, 32syl3anc 1372 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) (𝑆 𝑇))
3429, 33eqbrtrrd 5064 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) (𝑆 𝑇))
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem54 37395 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)
3621, 4atbase 36958 . . . . . . 7 (((𝐺 𝐻) 𝐵) ∈ 𝐴 → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
3735, 36syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
3821, 2, 6latlem12 17816 . . . . . 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 1373 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐺 𝐻) 𝐵) (𝑃 𝑄) ∧ ((𝐺 𝐻) 𝐵) (𝑆 𝑇)) ↔ ((𝐺 𝐻) 𝐵) ((𝑃 𝑄) (𝑆 𝑇))))
4028, 34, 39mpbi2and 712 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ((𝑃 𝑄) (𝑆 𝑇)))
41 dalem57.d . . . 4 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
4240, 41breqtrrdi 5082 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) 𝐷)
43 hlatl 37029 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
4418, 43syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
451, 2, 3, 4, 6, 7, 8, 9, 41dalemdea 37331 . . . . 5 (𝜑𝐷𝐴)
46453ad2ant1 1134 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐷𝐴)
472, 4atcmp 36980 . . . 4 ((𝐾 ∈ AtLat ∧ ((𝐺 𝐻) 𝐵) ∈ 𝐴𝐷𝐴) → (((𝐺 𝐻) 𝐵) 𝐷 ↔ ((𝐺 𝐻) 𝐵) = 𝐷))
4844, 35, 46, 47syl3anc 1372 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐵) 𝐷 ↔ ((𝐺 𝐻) 𝐵) = 𝐷))
4942, 48mpbid 235 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) = 𝐷)
50 eqid 2739 . . . . 5 (LLines‘𝐾) = (LLines‘𝐾)
511, 2, 3, 4, 5, 6, 50, 7, 8, 9, 10, 11, 12, 13dalem53 37394 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
5221, 50llnbase 37178 . . . 4 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
5351, 52syl 17 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
5421, 2, 6latmle2 17815 . . 3 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐵) 𝐵)
5516, 23, 53, 54syl3anc 1372 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) 𝐵)
5649, 55eqbrtrrd 5064 1 ((𝜑𝑌 = 𝑍𝜓) → 𝐷 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wne 2935   class class class wbr 5040  cfv 6349  (class class class)co 7182  Basecbs 16598  lecple 16687  joincjn 17682  meetcmee 17683  Latclat 17783  Atomscatm 36932  AtLatcal 36933  HLchlt 37019  LLinesclln 37160  LPlanesclpl 37161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7139  df-ov 7185  df-oprab 7186  df-proset 17666  df-poset 17684  df-plt 17696  df-lub 17712  df-glb 17713  df-join 17714  df-meet 17715  df-p0 17777  df-lat 17784  df-clat 17846  df-oposet 36845  df-ol 36847  df-oml 36848  df-covers 36935  df-ats 36936  df-atl 36967  df-cvlat 36991  df-hlat 37020  df-llines 37167  df-lplanes 37168  df-lvols 37169
This theorem is referenced by:  dalem58  37399  dalem60  37401
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