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Theorem dalem55 36730
 Description: Lemma for dath 36739. Lines 𝐺𝐻 and 𝑃𝑄 intersect at the auxiliary line 𝐵 (later shown to be an axis of perspectivity; see dalem60 36735). (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 36627 . . . . 5 (𝜑𝐾 ∈ Lat)
323ad2ant1 1127 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
41dalemkehl 36626 . . . . . 6 (𝜑𝐾 ∈ HL)
543ad2ant1 1127 . . . . 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 36699 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐺𝐴)
16 dalem54.h . . . . . 6 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
171, 6, 7, 8, 9, 10, 11, 12, 13, 16dalem29 36704 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐻𝐴)
18 eqid 2825 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
1918, 7, 8hlatjcl 36370 . . . . 5 ((𝐾 ∈ HL ∧ 𝐺𝐴𝐻𝐴) → (𝐺 𝐻) ∈ (Base‘𝐾))
205, 15, 17, 19syl3anc 1365 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ∈ (Base‘𝐾))
211, 7, 8dalempjqeb 36648 . . . . 5 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
22213ad2ant1 1127 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) ∈ (Base‘𝐾))
2318, 6, 10latmle1 17678 . . . 4 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻))
243, 20, 22, 23syl3anc 1365 . . 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 36709 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → 𝐼𝐴)
2718, 8atbase 36292 . . . . . . 7 (𝐼𝐴𝐼 ∈ (Base‘𝐾))
2826, 27syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝐼 ∈ (Base‘𝐾))
2918, 6, 7latlej1 17662 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → (𝐺 𝐻) ((𝐺 𝐻) 𝐼))
303, 20, 28, 29syl3anc 1365 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝐺 𝐻) ((𝐺 𝐻) 𝐼))
311, 8dalemreb 36644 . . . . . . . 8 (𝜑𝑅 ∈ (Base‘𝐾))
3218, 6, 7latlej1 17662 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑅))
332, 21, 31, 32syl3anc 1365 . . . . . . 7 (𝜑 → (𝑃 𝑄) ((𝑃 𝑄) 𝑅))
3433, 12breqtrrdi 5104 . . . . . 6 (𝜑 → (𝑃 𝑄) 𝑌)
35343ad2ant1 1127 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (𝑃 𝑄) 𝑌)
361, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25dalem42 36717 . . . . . . 7 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ 𝑂)
3718, 11lplnbase 36537 . . . . . . 7 (((𝐺 𝐻) 𝐼) ∈ 𝑂 → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
3836, 37syl 17 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐼) ∈ (Base‘𝐾))
391, 11dalemyeb 36652 . . . . . . 7 (𝜑𝑌 ∈ (Base‘𝐾))
40393ad2ant1 1127 . . . . . 6 ((𝜑𝑌 = 𝑍𝜓) → 𝑌 ∈ (Base‘𝐾))
4118, 6, 10latmlem12 17685 . . . . . 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 1373 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) ((𝐺 𝐻) 𝐼) ∧ (𝑃 𝑄) 𝑌) → ((𝐺 𝐻) (𝑃 𝑄)) (((𝐺 𝐻) 𝐼) 𝑌)))
4330, 35, 42mp2and 695 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) (((𝐺 𝐻) 𝐼) 𝑌))
44 dalem54.b1 . . . 4 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
4543, 44breqtrrdi 5104 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) 𝐵)
4618, 10latmcl 17654 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐺 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
473, 20, 22, 46syl3anc 1365 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ (Base‘𝐾))
48 eqid 2825 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
491, 6, 7, 8, 9, 10, 48, 11, 12, 13, 14, 16, 25, 44dalem53 36728 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
5018, 48llnbase 36512 . . . . 5 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
5149, 50syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
5218, 6, 10latlem12 17680 . . . 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 1366 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((((𝐺 𝐻) (𝑃 𝑄)) (𝐺 𝐻) ∧ ((𝐺 𝐻) (𝑃 𝑄)) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵)))
5424, 45, 53mpbi2and 708 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵))
55 hlatl 36363 . . . 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 36727 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)
581, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25, 44dalem54 36729 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)
596, 8atcmp 36314 . . 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 1365 . 2 ((𝜑𝑌 = 𝑍𝜓) → (((𝐺 𝐻) (𝑃 𝑄)) ((𝐺 𝐻) 𝐵) ↔ ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵)))
6154, 60mpbid 233 1 ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   ≠ wne 3020   class class class wbr 5062  ‘cfv 6351  (class class class)co 7151  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  Latclat 17647  Atomscatm 36266  AtLatcal 36267  HLchlt 36353  LLinesclln 36494  LPlanesclpl 36495 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36179  df-ol 36181  df-oml 36182  df-covers 36269  df-ats 36270  df-atl 36301  df-cvlat 36325  df-hlat 36354  df-llines 36501  df-lplanes 36502  df-lvols 36503 This theorem is referenced by:  dalem56  36731  dalem57  36732
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