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Theorem dalem55 39729
Description: Lemma for dath 39738. Lines 𝐺𝐻 and 𝑃𝑄 intersect at the auxiliary line 𝐵 (later shown to be an axis of perspectivity; see dalem60 39734). (Contributed by NM, 8-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))))
dalem54.m = (meet‘𝐾)
dalem54.o 𝑂 = (LPlanes‘𝐾)
dalem54.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem54.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem54.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
dalem54.h 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
dalem54.i 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
dalem54.b1 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
Assertion
Ref Expression
dalem55 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵))

Proof of Theorem dalem55
StepHypRef Expression
1 dalem.ph . . . . . 6 (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)))))
21dalemkelat 39626 . . . . 5 (𝜑𝐾 ∈ Lat)
323ad2ant1 1134 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
41dalemkehl 39625 . . . . . 6 (𝜑𝐾 ∈ HL)
543ad2ant1 1134 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
6 dalem.l . . . . . 6 = (le‘𝐾)
7 dalem.j . . . . . 6 = (join‘𝐾)
8 dalem.a . . . . . 6 𝐴 = (Atoms‘𝐾)
9 dalem.ps . . . . . 6 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
10 dalem54.m . . . . . 6 = (meet‘𝐾)
11 dalem54.o . . . . . 6 𝑂 = (LPlanes‘𝐾)
12 dalem54.y . . . . . 6 𝑌 = ((𝑃 𝑄) 𝑅)
13 dalem54.z . . . . . 6 𝑍 = ((𝑆 𝑇) 𝑈)
14 dalem54.g . . . . . 6 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
151, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem23 39698 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
16 dalem54.h . . . . . 6 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
171, 6, 7, 8, 9, 10, 11, 12, 13, 16dalem29 39703 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
18 eqid 2737 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
1918, 7, 8hlatjcl 39368 . . . . 5 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
205, 15, 17, 19syl3anc 1373 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
211, 7, 8dalempjqeb 39647 . . . . 5 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
22213ad2ant1 1134 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
2318, 6, 10latmle1 18509 . . . 4 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻))
243, 20, 22, 23syl3anc 1373 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻))
25 dalem54.i . . . . . . . 8 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
261, 6, 7, 8, 9, 10, 11, 12, 13, 25dalem34 39708 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2718, 8atbase 39290 . . . . . . 7 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
2826, 27syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
2918, 6, 7latlej1 18493 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → (𝐺 𝐻) ((𝐺 𝐻) 𝐼))
303, 20, 28, 29syl3anc 1373 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ((𝐺 𝐻) 𝐼))
311, 8dalemreb 39643 . . . . . . . 8 (𝜑𝑅 ∈ (Base‘𝐾))
3218, 6, 7latlej1 18493 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑅))
332, 21, 31, 32syl3anc 1373 . . . . . . 7 (𝜑 → (𝑃 𝑄) ((𝑃 𝑄) 𝑅))
3433, 12breqtrrdi 5185 . . . . . 6 (𝜑 → (𝑃 𝑄) 𝑌)
35343ad2ant1 1134 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) 𝑌)
361, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25dalem42 39716 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
3718, 11lplnbase 39536 . . . . . . 7 (((𝐺 𝐻) 𝐼) ∈ 𝑂 → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
3836, 37syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
391, 11dalemyeb 39651 . . . . . . 7 (𝜑𝑌 ∈ (Base‘𝐾))
40393ad2ant1 1134 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
4118, 6, 10latmlem12 18516 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝐺 𝐻) ∈ (Base‘𝐾) ∧ ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾)) ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → (((𝐺 𝐻) ((𝐺 𝐻) 𝐼) ∧ (𝑃 𝑄) 𝑌) → ((𝐺 𝐻) (𝑃 𝑄)) (((𝐺 𝐻) 𝐼) 𝑌)))
423, 20, 38, 22, 40, 41syl122anc 1381 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) ((𝐺 𝐻) 𝐼) ∧ (𝑃 𝑄) 𝑌) → ((𝐺 𝐻) (𝑃 𝑄)) (((𝐺 𝐻) 𝐼) 𝑌)))
4330, 35, 42mp2and 699 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (((𝐺 𝐻) 𝐼) 𝑌))
44 dalem54.b1 . . . 4 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
4543, 44breqtrrdi 5185 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) 𝐵)
4618, 10latmcl 18485 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
473, 20, 22, 46syl3anc 1373 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
48 eqid 2737 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
491, 6, 7, 8, 9, 10, 48, 11, 12, 13, 14, 16, 25, 44dalem53 39727 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
5018, 48llnbase 39511 . . . . 5 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
5149, 50syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
5218, 6, 10latlem12 18511 . . . 4 ((𝐾 ∈ Lat ∧ (((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾) ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾))) → ((((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻) ∧ ((𝐺 𝐻) (𝑃 𝑄)) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵)))
533, 47, 20, 51, 52syl13anc 1374 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻) ∧ ((𝐺 𝐻) (𝑃 𝑄)) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵)))
5424, 45, 53mpbi2and 712 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵))
55 hlatl 39361 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
565, 55syl 17 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ AtLat)
571, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25dalem52 39726 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)
581, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25, 44dalem54 39728 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)
596, 8atcmp 39312 . . 3 ((𝐾 ∈ AtLat ∧ ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴 ∧ ((𝐺 𝐻) 𝐵) ∈ 𝐴) → (((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵)))
6056, 57, 58, 59syl3anc 1373 . 2 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵)))
6154, 60mpbid 232 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  Latclat 18476  Atomscatm 39264  AtLatcal 39265  HLchlt 39351  LLinesclln 39493  LPlanesclpl 39494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lvols 39502
This theorem is referenced by:  dalem56  39730  dalem57  39731
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