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Theorem dalem55 34528
 Description: Lemma for dath 34537. Lines 𝐺𝐻 and 𝑃𝑄 intersect at the auxiliary line 𝐵 (later shown to be an axis of perspectivity; see dalem60 34533). (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 34425 . . . . 5 (𝜑𝐾 ∈ Lat)
323ad2ant1 1080 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
41dalemkehl 34424 . . . . . 6 (𝜑𝐾 ∈ HL)
543ad2ant1 1080 . . . . 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 34497 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
16 dalem54.h . . . . . 6 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
171, 6, 7, 8, 9, 10, 11, 12, 13, 16dalem29 34502 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
18 eqid 2621 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
1918, 7, 8hlatjcl 34168 . . . . 5 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
205, 15, 17, 19syl3anc 1323 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
211, 7, 8dalempjqeb 34446 . . . . 5 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
22213ad2ant1 1080 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
2318, 6, 10latmle1 17008 . . . 4 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻))
243, 20, 22, 23syl3anc 1323 . . 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 34507 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2718, 8atbase 34091 . . . . . . 7 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
2826, 27syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
2918, 6, 7latlej1 16992 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → (𝐺 𝐻) ((𝐺 𝐻) 𝐼))
303, 20, 28, 29syl3anc 1323 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ((𝐺 𝐻) 𝐼))
311, 8dalemreb 34442 . . . . . . . 8 (𝜑𝑅 ∈ (Base‘𝐾))
3218, 6, 7latlej1 16992 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑅))
332, 21, 31, 32syl3anc 1323 . . . . . . 7 (𝜑 → (𝑃 𝑄) ((𝑃 𝑄) 𝑅))
3433, 12syl6breqr 4660 . . . . . 6 (𝜑 → (𝑃 𝑄) 𝑌)
35343ad2ant1 1080 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) 𝑌)
361, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25dalem42 34515 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
3718, 11lplnbase 34335 . . . . . . 7 (((𝐺 𝐻) 𝐼) ∈ 𝑂 → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
3836, 37syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
391, 11dalemyeb 34450 . . . . . . 7 (𝜑𝑌 ∈ (Base‘𝐾))
40393ad2ant1 1080 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
4118, 6, 10latmlem12 17015 . . . . . 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 1332 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) ((𝐺 𝐻) 𝐼) ∧ (𝑃 𝑄) 𝑌) → ((𝐺 𝐻) (𝑃 𝑄)) (((𝐺 𝐻) 𝐼) 𝑌)))
4330, 35, 42mp2and 714 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (((𝐺 𝐻) 𝐼) 𝑌))
44 dalem54.b1 . . . 4 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
4543, 44syl6breqr 4660 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) 𝐵)
4618, 10latmcl 16984 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
473, 20, 22, 46syl3anc 1323 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
48 eqid 2621 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
491, 6, 7, 8, 9, 10, 48, 11, 12, 13, 14, 16, 25, 44dalem53 34526 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
5018, 48llnbase 34310 . . . . 5 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
5149, 50syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
5218, 6, 10latlem12 17010 . . . 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 1325 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻) ∧ ((𝐺 𝐻) (𝑃 𝑄)) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵)))
5424, 45, 53mpbi2and 955 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵))
55 hlatl 34162 . . . 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 34525 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)
581, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25, 44dalem54 34527 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)
596, 8atcmp 34113 . . 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 1323 . 2 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵)))
6154, 60mpbid 222 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790   class class class wbr 4618  ‘cfv 5852  (class class class)co 6610  Basecbs 15792  lecple 15880  joincjn 16876  meetcmee 16877  Latclat 16977  Atomscatm 34065  AtLatcal 34066  HLchlt 34152  LLinesclln 34292  LPlanesclpl 34293 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-preset 16860  df-poset 16878  df-plt 16890  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-p0 16971  df-lat 16978  df-clat 17040  df-oposet 33978  df-ol 33980  df-oml 33981  df-covers 34068  df-ats 34069  df-atl 34100  df-cvlat 34124  df-hlat 34153  df-llines 34299  df-lplanes 34300  df-lvols 34301 This theorem is referenced by:  dalem56  34529  dalem57  34530
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