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Theorem dalem55 35803
Description: Lemma for dath 35812. Lines 𝐺𝐻 and 𝑃𝑄 intersect at the auxiliary line 𝐵 (later shown to be an axis of perspectivity; see dalem60 35808). (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 35700 . . . . 5 (𝜑𝐾 ∈ Lat)
323ad2ant1 1169 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
41dalemkehl 35699 . . . . . 6 (𝜑𝐾 ∈ HL)
543ad2ant1 1169 . . . . 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 35772 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
16 dalem54.h . . . . . 6 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
171, 6, 7, 8, 9, 10, 11, 12, 13, 16dalem29 35777 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
18 eqid 2826 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
1918, 7, 8hlatjcl 35443 . . . . 5 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
205, 15, 17, 19syl3anc 1496 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
211, 7, 8dalempjqeb 35721 . . . . 5 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
22213ad2ant1 1169 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
2318, 6, 10latmle1 17430 . . . 4 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻))
243, 20, 22, 23syl3anc 1496 . . 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 35782 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2718, 8atbase 35365 . . . . . . 7 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
2826, 27syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
2918, 6, 7latlej1 17414 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → (𝐺 𝐻) ((𝐺 𝐻) 𝐼))
303, 20, 28, 29syl3anc 1496 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ((𝐺 𝐻) 𝐼))
311, 8dalemreb 35717 . . . . . . . 8 (𝜑𝑅 ∈ (Base‘𝐾))
3218, 6, 7latlej1 17414 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑅))
332, 21, 31, 32syl3anc 1496 . . . . . . 7 (𝜑 → (𝑃 𝑄) ((𝑃 𝑄) 𝑅))
3433, 12syl6breqr 4916 . . . . . 6 (𝜑 → (𝑃 𝑄) 𝑌)
35343ad2ant1 1169 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) 𝑌)
361, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25dalem42 35790 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
3718, 11lplnbase 35610 . . . . . . 7 (((𝐺 𝐻) 𝐼) ∈ 𝑂 → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
3836, 37syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
391, 11dalemyeb 35725 . . . . . . 7 (𝜑𝑌 ∈ (Base‘𝐾))
40393ad2ant1 1169 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
4118, 6, 10latmlem12 17437 . . . . . 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 1504 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) ((𝐺 𝐻) 𝐼) ∧ (𝑃 𝑄) 𝑌) → ((𝐺 𝐻) (𝑃 𝑄)) (((𝐺 𝐻) 𝐼) 𝑌)))
4330, 35, 42mp2and 692 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (((𝐺 𝐻) 𝐼) 𝑌))
44 dalem54.b1 . . . 4 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
4543, 44syl6breqr 4916 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) 𝐵)
4618, 10latmcl 17406 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
473, 20, 22, 46syl3anc 1496 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
48 eqid 2826 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
491, 6, 7, 8, 9, 10, 48, 11, 12, 13, 14, 16, 25, 44dalem53 35801 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
5018, 48llnbase 35585 . . . . 5 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
5149, 50syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
5218, 6, 10latlem12 17432 . . . 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 1497 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻) ∧ ((𝐺 𝐻) (𝑃 𝑄)) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵)))
5424, 45, 53mpbi2and 705 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵))
55 hlatl 35436 . . . 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 35800 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)
581, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25, 44dalem54 35802 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)
596, 8atcmp 35387 . . 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 1496 . 2 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵)))
6154, 60mpbid 224 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wne 3000   class class class wbr 4874  cfv 6124  (class class class)co 6906  Basecbs 16223  lecple 16313  joincjn 17298  meetcmee 17299  Latclat 17399  Atomscatm 35339  AtLatcal 35340  HLchlt 35426  LLinesclln 35567  LPlanesclpl 35568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-proset 17282  df-poset 17300  df-plt 17312  df-lub 17328  df-glb 17329  df-join 17330  df-meet 17331  df-p0 17393  df-lat 17400  df-clat 17462  df-oposet 35252  df-ol 35254  df-oml 35255  df-covers 35342  df-ats 35343  df-atl 35374  df-cvlat 35398  df-hlat 35427  df-llines 35574  df-lplanes 35575  df-lvols 35576
This theorem is referenced by:  dalem56  35804  dalem57  35805
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