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Theorem dalem12 36853
Description: Lemma for dath 36914. Analogue of dalem10 36851 for 𝐹. (Contributed by NM, 11-Aug-2012.)
Hypotheses
Ref Expression
dalema.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)))))
dalemc.l = (le‘𝐾)
dalemc.j = (join‘𝐾)
dalemc.a 𝐴 = (Atoms‘𝐾)
dalem12.m = (meet‘𝐾)
dalem12.o 𝑂 = (LPlanes‘𝐾)
dalem12.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem12.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem12.x 𝑋 = (𝑌 𝑍)
dalem12.f 𝐹 = ((𝑅 𝑃) (𝑈 𝑆))
Assertion
Ref Expression
dalem12 (𝜑𝐹 𝑋)

Proof of Theorem dalem12
StepHypRef Expression
1 dalema.ph . . . 4 (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 dalemc.l . . . 4 = (le‘𝐾)
3 dalemc.j . . . 4 = (join‘𝐾)
4 dalemc.a . . . 4 𝐴 = (Atoms‘𝐾)
5 dalem12.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
6 dalem12.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
71, 2, 3, 4, 5, 6dalemrot 36835 . . 3 (𝜑 → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))))
8 biid 264 . . . 4 ((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))) ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))))
9 dalem12.m . . . 4 = (meet‘𝐾)
10 dalem12.o . . . 4 𝑂 = (LPlanes‘𝐾)
11 eqid 2821 . . . 4 ((𝑄 𝑅) 𝑃) = ((𝑄 𝑅) 𝑃)
12 eqid 2821 . . . 4 ((𝑇 𝑈) 𝑆) = ((𝑇 𝑈) 𝑆)
13 eqid 2821 . . . 4 (((𝑄 𝑅) 𝑃) ((𝑇 𝑈) 𝑆)) = (((𝑄 𝑅) 𝑃) ((𝑇 𝑈) 𝑆))
14 dalem12.f . . . 4 𝐹 = ((𝑅 𝑃) (𝑈 𝑆))
158, 2, 3, 4, 9, 10, 11, 12, 13, 14dalem11 36852 . . 3 ((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) ∧ (((𝑄 𝑅) 𝑃) ∈ 𝑂 ∧ ((𝑇 𝑈) 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃) ∧ ¬ 𝐶 (𝑃 𝑄)) ∧ (¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆) ∧ ¬ 𝐶 (𝑆 𝑇)) ∧ (𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈) ∧ 𝐶 (𝑃 𝑆)))) → 𝐹 (((𝑄 𝑅) 𝑃) ((𝑇 𝑈) 𝑆)))
167, 15syl 17 . 2 (𝜑𝐹 (((𝑄 𝑅) 𝑃) ((𝑇 𝑈) 𝑆)))
17 dalem12.x . . 3 𝑋 = (𝑌 𝑍)
181, 3, 4dalemqrprot 36826 . . . . 5 (𝜑 → ((𝑄 𝑅) 𝑃) = ((𝑃 𝑄) 𝑅))
1918, 5syl6reqr 2875 . . . 4 (𝜑𝑌 = ((𝑄 𝑅) 𝑃))
201dalemkehl 36801 . . . . . 6 (𝜑𝐾 ∈ HL)
211dalemtea 36808 . . . . . 6 (𝜑𝑇𝐴)
221dalemuea 36809 . . . . . 6 (𝜑𝑈𝐴)
231dalemsea 36807 . . . . . 6 (𝜑𝑆𝐴)
243, 4hlatjrot 36551 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑇𝐴𝑈𝐴𝑆𝐴)) → ((𝑇 𝑈) 𝑆) = ((𝑆 𝑇) 𝑈))
2520, 21, 22, 23, 24syl13anc 1369 . . . . 5 (𝜑 → ((𝑇 𝑈) 𝑆) = ((𝑆 𝑇) 𝑈))
2625, 6syl6reqr 2875 . . . 4 (𝜑𝑍 = ((𝑇 𝑈) 𝑆))
2719, 26oveq12d 7148 . . 3 (𝜑 → (𝑌 𝑍) = (((𝑄 𝑅) 𝑃) ((𝑇 𝑈) 𝑆)))
2817, 27syl5eq 2868 . 2 (𝜑𝑋 = (((𝑄 𝑅) 𝑃) ((𝑇 𝑈) 𝑆)))
2916, 28breqtrrd 5067 1 (𝜑𝐹 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115   class class class wbr 5039  cfv 6328  (class class class)co 7130  Basecbs 16462  lecple 16551  joincjn 17533  meetcmee 17534  Atomscatm 36441  HLchlt 36528  LPlanesclpl 36670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-proset 17517  df-poset 17535  df-lub 17563  df-glb 17564  df-join 17565  df-meet 17566  df-lat 17635  df-ats 36445  df-atl 36476  df-cvlat 36500  df-hlat 36529  df-lplanes 36677
This theorem is referenced by:  dalem16  36857
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