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Theorem dalem10 36969
 Description: Lemma for dath 37032. Atom 𝐷 belongs to the axis of perspectivity 𝑋. (Contributed by NM, 19-Jul-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)
dalem10.m = (meet‘𝐾)
dalem10.o 𝑂 = (LPlanes‘𝐾)
dalem10.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem10.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem10.x 𝑋 = (𝑌 𝑍)
dalem10.d 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
Assertion
Ref Expression
dalem10 (𝜑𝐷 𝑋)

Proof of Theorem dalem10
StepHypRef Expression
1 dalema.ph . . . . 5 (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 36920 . . . 4 (𝜑𝐾 ∈ Lat)
3 dalemc.j . . . . 5 = (join‘𝐾)
4 dalemc.a . . . . 5 𝐴 = (Atoms‘𝐾)
51, 3, 4dalempjqeb 36941 . . . 4 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
61, 4dalemreb 36937 . . . 4 (𝜑𝑅 ∈ (Base‘𝐾))
7 eqid 2798 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
8 dalemc.l . . . . 5 = (le‘𝐾)
97, 8, 3latlej1 17662 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 𝑄) ((𝑃 𝑄) 𝑅))
102, 5, 6, 9syl3anc 1368 . . 3 (𝜑 → (𝑃 𝑄) ((𝑃 𝑄) 𝑅))
111, 3, 4dalemsjteb 36942 . . . 4 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
121, 4dalemueb 36940 . . . 4 (𝜑𝑈 ∈ (Base‘𝐾))
137, 8, 3latlej1 17662 . . . 4 ((𝐾 ∈ Lat ∧ (𝑆 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑆 𝑇) ((𝑆 𝑇) 𝑈))
142, 11, 12, 13syl3anc 1368 . . 3 (𝜑 → (𝑆 𝑇) ((𝑆 𝑇) 𝑈))
15 dalem10.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
16 dalem10.o . . . . . 6 𝑂 = (LPlanes‘𝐾)
171, 16dalemyeb 36945 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐾))
1815, 17eqeltrrid 2895 . . . 4 (𝜑 → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
19 dalem10.z . . . . 5 𝑍 = ((𝑆 𝑇) 𝑈)
201dalemzeo 36929 . . . . . 6 (𝜑𝑍𝑂)
217, 16lplnbase 36830 . . . . . 6 (𝑍𝑂𝑍 ∈ (Base‘𝐾))
2220, 21syl 17 . . . . 5 (𝜑𝑍 ∈ (Base‘𝐾))
2319, 22eqeltrrid 2895 . . . 4 (𝜑 → ((𝑆 𝑇) 𝑈) ∈ (Base‘𝐾))
24 dalem10.m . . . . 5 = (meet‘𝐾)
257, 8, 24latmlem12 17685 . . . 4 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾)) ∧ ((𝑆 𝑇) ∈ (Base‘𝐾) ∧ ((𝑆 𝑇) 𝑈) ∈ (Base‘𝐾))) → (((𝑃 𝑄) ((𝑃 𝑄) 𝑅) ∧ (𝑆 𝑇) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈))))
262, 5, 18, 11, 23, 25syl122anc 1376 . . 3 (𝜑 → (((𝑃 𝑄) ((𝑃 𝑄) 𝑅) ∧ (𝑆 𝑇) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈))))
2710, 14, 26mp2and 698 . 2 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
28 dalem10.d . 2 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
29 dalem10.x . . 3 𝑋 = (𝑌 𝑍)
3015, 19oveq12i 7147 . . 3 (𝑌 𝑍) = (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈))
3129, 30eqtri 2821 . 2 𝑋 = (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈))
3227, 28, 313brtr4g 5064 1 (𝜑𝐷 𝑋)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  Latclat 17647  Atomscatm 36559  HLchlt 36646  LPlanesclpl 36788 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-poset 17548  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-lat 17648  df-ats 36563  df-atl 36594  df-cvlat 36618  df-hlat 36647  df-lplanes 36795 This theorem is referenced by:  dalem11  36970  dalem16  36975  dalem54  37022
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