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Theorem dalem8 36673
Description: Lemma for dath 36739. 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 36671 . . . 4 (𝜑𝑆 𝑊)
102, 3, 4, 5, 6, 7, 1, 8dalem7 36672 . . . 4 (𝜑𝑇 𝑊)
112dalemkelat 36627 . . . . 5 (𝜑𝐾 ∈ Lat)
122, 5dalemseb 36645 . . . . 5 (𝜑𝑆 ∈ (Base‘𝐾))
132, 5dalemteb 36646 . . . . 5 (𝜑𝑇 ∈ (Base‘𝐾))
142, 6dalemyeb 36652 . . . . . . 7 (𝜑𝑌 ∈ (Base‘𝐾))
152, 5dalemceb 36641 . . . . . . 7 (𝜑𝐶 ∈ (Base‘𝐾))
16 eqid 2826 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
1716, 4latjcl 17651 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑌 𝐶) ∈ (Base‘𝐾))
1811, 14, 15, 17syl3anc 1365 . . . . . 6 (𝜑 → (𝑌 𝐶) ∈ (Base‘𝐾))
198, 18eqeltrid 2922 . . . . 5 (𝜑𝑊 ∈ (Base‘𝐾))
2016, 3, 4latjle12 17662 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑆 𝑊𝑇 𝑊) ↔ (𝑆 𝑇) 𝑊))
2111, 12, 13, 19, 20syl13anc 1366 . . . 4 (𝜑 → ((𝑆 𝑊𝑇 𝑊) ↔ (𝑆 𝑇) 𝑊))
229, 10, 21mpbi2and 708 . . 3 (𝜑 → (𝑆 𝑇) 𝑊)
232, 3, 4, 5, 6, 7, 8dalem5 36670 . . 3 (𝜑𝑈 𝑊)
242, 4, 5dalemsjteb 36649 . . . 4 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
252, 5dalemueb 36647 . . . 4 (𝜑𝑈 ∈ (Base‘𝐾))
2616, 3, 4latjle12 17662 . . . 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 1366 . . 3 (𝜑 → (((𝑆 𝑇) 𝑊𝑈 𝑊) ↔ ((𝑆 𝑇) 𝑈) 𝑊))
2822, 23, 27mpbi2and 708 . 2 (𝜑 → ((𝑆 𝑇) 𝑈) 𝑊)
291, 28eqbrtrid 5098 1 (𝜑𝑍 𝑊)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107   class class class wbr 5063  cfv 6352  (class class class)co 7148  Basecbs 16473  lecple 16562  joincjn 17544  Latclat 17645  Atomscatm 36266  HLchlt 36353  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 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-proset 17528  df-poset 17546  df-plt 17558  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-p0 17639  df-lat 17646  df-clat 17708  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
This theorem is referenced by:  dalem13  36679
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