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Theorem dalem57 37743
Description: Lemma for dath 37750. 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 37741 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵))
151dalemkelat 37638 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
16153ad2ant1 1132 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
171dalemkehl 37637 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
18173ad2ant1 1132 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10dalem23 37710 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
201, 2, 3, 4, 5, 6, 7, 8, 9, 11dalem29 37715 . . . . . . . 8 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
21 eqid 2738 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
2221, 3, 4hlatjcl 37381 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
2318, 19, 20, 22syl3anc 1370 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
241, 3, 4dalempjqeb 37659 . . . . . . . 8 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
25243ad2ant1 1132 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
2621, 2, 6latmle2 18183 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) (𝑃 𝑄))
2716, 23, 25, 26syl3anc 1370 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (𝑃 𝑄))
2814, 27eqbrtrrd 5098 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) (𝑃 𝑄))
291, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem56 37742 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) = ((𝐺 𝐻) 𝐵))
301, 3, 4dalemsjteb 37660 . . . . . . . 8 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
31303ad2ant1 1132 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → (𝑆 𝑇) ∈ (Base‘𝐾))
3221, 2, 6latmle2 18183 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑆 𝑇)) (𝑆 𝑇))
3316, 23, 31, 32syl3anc 1370 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) (𝑆 𝑇))
3429, 33eqbrtrrd 5098 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) (𝑆 𝑇))
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dalem54 37740 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)
3621, 4atbase 37303 . . . . . . 7 (((𝐺 𝐻) 𝐵) ∈ 𝐴 → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
3735, 36syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ (Base‘𝐾))
3821, 2, 6latlem12 18184 . . . . . 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 1371 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐺 𝐻) 𝐵) (𝑃 𝑄) ∧ ((𝐺 𝐻) 𝐵) (𝑆 𝑇)) ↔ ((𝐺 𝐻) 𝐵) ((𝑃 𝑄) (𝑆 𝑇))))
4028, 34, 39mpbi2and 709 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ((𝑃 𝑄) (𝑆 𝑇)))
41 dalem57.d . . . 4 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
4240, 41breqtrrdi 5116 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) 𝐷)
43 hlatl 37374 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
4418, 43syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
451, 2, 3, 4, 6, 7, 8, 9, 41dalemdea 37676 . . . . 5 (𝜑𝐷𝐴)
46453ad2ant1 1132 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐷𝐴)
472, 4atcmp 37325 . . . 4 ((𝐾 ∈ AtLat ∧ ((𝐺 𝐻) 𝐵) ∈ 𝐴𝐷𝐴) → (((𝐺 𝐻) 𝐵) 𝐷 ↔ ((𝐺 𝐻) 𝐵) = 𝐷))
4844, 35, 46, 47syl3anc 1370 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) 𝐵) 𝐷 ↔ ((𝐺 𝐻) 𝐵) = 𝐷))
4942, 48mpbid 231 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) = 𝐷)
50 eqid 2738 . . . . 5 (LLines‘𝐾) = (LLines‘𝐾)
511, 2, 3, 4, 5, 6, 50, 7, 8, 9, 10, 11, 12, 13dalem53 37739 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
5221, 50llnbase 37523 . . . 4 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
5351, 52syl 17 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
5421, 2, 6latmle2 18183 . . 3 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → ((𝐺 𝐻) 𝐵) 𝐵)
5516, 23, 53, 54syl3anc 1370 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) 𝐵)
5649, 55eqbrtrrd 5098 1 ((𝜑𝑌 = 𝑍𝜓) → 𝐷 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943   class class class wbr 5074  cfv 6433  (class class class)co 7275  Basecbs 16912  lecple 16969  joincjn 18029  meetcmee 18030  Latclat 18149  Atomscatm 37277  AtLatcal 37278  HLchlt 37364  LLinesclln 37505  LPlanesclpl 37506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-lat 18150  df-clat 18217  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-llines 37512  df-lplanes 37513  df-lvols 37514
This theorem is referenced by:  dalem58  37744  dalem60  37746
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