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Theorem dalem8 35469
Description: Lemma for dath 35535. Plane 𝑍 belongs to the 3-dimensional space. (Contributed by NM, 21-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)
dalem6.o 𝑂 = (LPlanes‘𝐾)
dalem6.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem6.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem6.w 𝑊 = (𝑌 𝐶)
Assertion
Ref Expression
dalem8 (𝜑𝑍 𝑊)

Proof of Theorem dalem8
StepHypRef Expression
1 dalem6.z . 2 𝑍 = ((𝑆 𝑇) 𝑈)
2 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)))))
3 dalemc.l . . . . 5 = (le‘𝐾)
4 dalemc.j . . . . 5 = (join‘𝐾)
5 dalemc.a . . . . 5 𝐴 = (Atoms‘𝐾)
6 dalem6.o . . . . 5 𝑂 = (LPlanes‘𝐾)
7 dalem6.y . . . . 5 𝑌 = ((𝑃 𝑄) 𝑅)
8 dalem6.w . . . . 5 𝑊 = (𝑌 𝐶)
92, 3, 4, 5, 6, 7, 1, 8dalem6 35467 . . . 4 (𝜑𝑆 𝑊)
102, 3, 4, 5, 6, 7, 1, 8dalem7 35468 . . . 4 (𝜑𝑇 𝑊)
112dalemkelat 35423 . . . . 5 (𝜑𝐾 ∈ Lat)
122, 5dalemseb 35441 . . . . 5 (𝜑𝑆 ∈ (Base‘𝐾))
132, 5dalemteb 35442 . . . . 5 (𝜑𝑇 ∈ (Base‘𝐾))
142, 6dalemyeb 35448 . . . . . . 7 (𝜑𝑌 ∈ (Base‘𝐾))
152, 5dalemceb 35437 . . . . . . 7 (𝜑𝐶 ∈ (Base‘𝐾))
16 eqid 2817 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
1716, 4latjcl 17276 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑌 𝐶) ∈ (Base‘𝐾))
1811, 14, 15, 17syl3anc 1483 . . . . . 6 (𝜑 → (𝑌 𝐶) ∈ (Base‘𝐾))
198, 18syl5eqel 2900 . . . . 5 (𝜑𝑊 ∈ (Base‘𝐾))
2016, 3, 4latjle12 17287 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑆 𝑊𝑇 𝑊) ↔ (𝑆 𝑇) 𝑊))
2111, 12, 13, 19, 20syl13anc 1484 . . . 4 (𝜑 → ((𝑆 𝑊𝑇 𝑊) ↔ (𝑆 𝑇) 𝑊))
229, 10, 21mpbi2and 694 . . 3 (𝜑 → (𝑆 𝑇) 𝑊)
232, 3, 4, 5, 6, 7, 8dalem5 35466 . . 3 (𝜑𝑈 𝑊)
242, 4, 5dalemsjteb 35445 . . . 4 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
252, 5dalemueb 35443 . . . 4 (𝜑𝑈 ∈ (Base‘𝐾))
2616, 3, 4latjle12 17287 . . . 4 ((𝐾 ∈ Lat ∧ ((𝑆 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑆 𝑇) 𝑊𝑈 𝑊) ↔ ((𝑆 𝑇) 𝑈) 𝑊))
2711, 24, 25, 19, 26syl13anc 1484 . . 3 (𝜑 → (((𝑆 𝑇) 𝑊𝑈 𝑊) ↔ ((𝑆 𝑇) 𝑈) 𝑊))
2822, 23, 27mpbi2and 694 . 2 (𝜑 → ((𝑆 𝑇) 𝑈) 𝑊)
291, 28syl5eqbr 4890 1 (𝜑𝑍 𝑊)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157   class class class wbr 4855  cfv 6111  (class class class)co 6884  Basecbs 16088  lecple 16180  joincjn 17169  Latclat 17270  Atomscatm 35062  HLchlt 35149  LPlanesclpl 35291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-riota 6845  df-ov 6887  df-oprab 6888  df-proset 17153  df-poset 17171  df-plt 17183  df-lub 17199  df-glb 17200  df-join 17201  df-meet 17202  df-p0 17264  df-lat 17271  df-clat 17333  df-oposet 34975  df-ol 34977  df-oml 34978  df-covers 35065  df-ats 35066  df-atl 35097  df-cvlat 35121  df-hlat 35150  df-llines 35297  df-lplanes 35298
This theorem is referenced by:  dalem13  35475
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