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Theorem dalem4 35553
Description: Lemma for dalemdnee 35554. (Contributed by NM, 10-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)
dalem3.m = (meet‘𝐾)
dalem3.o 𝑂 = (LPlanes‘𝐾)
dalem3.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem3.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem3.d 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
dalem3.e 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
Assertion
Ref Expression
dalem4 ((𝜑𝐷𝑇) → 𝐷𝐸)

Proof of Theorem dalem4
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)))))
2 dalemc.l . . . . 5 = (le‘𝐾)
3 dalemc.j . . . . 5 = (join‘𝐾)
4 dalemc.a . . . . 5 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4dalemswapyz 35544 . . . 4 (𝜑 → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (𝑍𝑂𝑌𝑂) ∧ ((¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (𝐶 (𝑆 𝑃) ∧ 𝐶 (𝑇 𝑄) ∧ 𝐶 (𝑈 𝑅)))))
65adantr 472 . . 3 ((𝜑𝐷𝑇) → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (𝑍𝑂𝑌𝑂) ∧ ((¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (𝐶 (𝑆 𝑃) ∧ 𝐶 (𝑇 𝑄) ∧ 𝐶 (𝑈 𝑅)))))
7 dalem3.d . . . . . 6 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
81dalemkelat 35512 . . . . . . 7 (𝜑𝐾 ∈ Lat)
91, 3, 4dalempjqeb 35533 . . . . . . 7 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
101, 3, 4dalemsjteb 35534 . . . . . . 7 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
11 eqid 2765 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
12 dalem3.m . . . . . . . 8 = (meet‘𝐾)
1311, 12latmcom 17343 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑆 𝑇) (𝑃 𝑄)))
148, 9, 10, 13syl3anc 1490 . . . . . 6 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑆 𝑇) (𝑃 𝑄)))
157, 14syl5eq 2811 . . . . 5 (𝜑𝐷 = ((𝑆 𝑇) (𝑃 𝑄)))
1615neeq1d 2996 . . . 4 (𝜑 → (𝐷𝑇 ↔ ((𝑆 𝑇) (𝑃 𝑄)) ≠ 𝑇))
1716biimpa 468 . . 3 ((𝜑𝐷𝑇) → ((𝑆 𝑇) (𝑃 𝑄)) ≠ 𝑇)
18 biid 252 . . . 4 ((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (𝑍𝑂𝑌𝑂) ∧ ((¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (𝐶 (𝑆 𝑃) ∧ 𝐶 (𝑇 𝑄) ∧ 𝐶 (𝑈 𝑅)))) ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (𝑍𝑂𝑌𝑂) ∧ ((¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (𝐶 (𝑆 𝑃) ∧ 𝐶 (𝑇 𝑄) ∧ 𝐶 (𝑈 𝑅)))))
19 dalem3.o . . . 4 𝑂 = (LPlanes‘𝐾)
20 dalem3.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
21 dalem3.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
22 eqid 2765 . . . 4 ((𝑆 𝑇) (𝑃 𝑄)) = ((𝑆 𝑇) (𝑃 𝑄))
23 eqid 2765 . . . 4 ((𝑇 𝑈) (𝑄 𝑅)) = ((𝑇 𝑈) (𝑄 𝑅))
2418, 2, 3, 4, 12, 19, 20, 21, 22, 23dalem3 35552 . . 3 (((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (𝑍𝑂𝑌𝑂) ∧ ((¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (𝐶 (𝑆 𝑃) ∧ 𝐶 (𝑇 𝑄) ∧ 𝐶 (𝑈 𝑅)))) ∧ ((𝑆 𝑇) (𝑃 𝑄)) ≠ 𝑇) → ((𝑆 𝑇) (𝑃 𝑄)) ≠ ((𝑇 𝑈) (𝑄 𝑅)))
256, 17, 24syl2anc 579 . 2 ((𝜑𝐷𝑇) → ((𝑆 𝑇) (𝑃 𝑄)) ≠ ((𝑇 𝑈) (𝑄 𝑅)))
2615adantr 472 . 2 ((𝜑𝐷𝑇) → 𝐷 = ((𝑆 𝑇) (𝑃 𝑄)))
27 dalem3.e . . . 4 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
281dalemkehl 35511 . . . . . 6 (𝜑𝐾 ∈ HL)
291dalemqea 35515 . . . . . 6 (𝜑𝑄𝐴)
301dalemrea 35516 . . . . . 6 (𝜑𝑅𝐴)
3111, 3, 4hlatjcl 35255 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
3228, 29, 30, 31syl3anc 1490 . . . . 5 (𝜑 → (𝑄 𝑅) ∈ (Base‘𝐾))
331, 3, 4dalemtjueb 35535 . . . . 5 (𝜑 → (𝑇 𝑈) ∈ (Base‘𝐾))
3411, 12latmcom 17343 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → ((𝑄 𝑅) (𝑇 𝑈)) = ((𝑇 𝑈) (𝑄 𝑅)))
358, 32, 33, 34syl3anc 1490 . . . 4 (𝜑 → ((𝑄 𝑅) (𝑇 𝑈)) = ((𝑇 𝑈) (𝑄 𝑅)))
3627, 35syl5eq 2811 . . 3 (𝜑𝐸 = ((𝑇 𝑈) (𝑄 𝑅)))
3736adantr 472 . 2 ((𝜑𝐷𝑇) → 𝐸 = ((𝑇 𝑈) (𝑄 𝑅)))
3825, 26, 373netr4d 3014 1 ((𝜑𝐷𝑇) → 𝐷𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937   class class class wbr 4809  cfv 6068  (class class class)co 6842  Basecbs 16132  lecple 16223  joincjn 17212  meetcmee 17213  Latclat 17313  Atomscatm 35151  HLchlt 35238  LPlanesclpl 35380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-proset 17196  df-poset 17214  df-plt 17226  df-lub 17242  df-glb 17243  df-join 17244  df-meet 17245  df-p0 17307  df-lat 17314  df-clat 17376  df-oposet 35064  df-ol 35066  df-oml 35067  df-covers 35154  df-ats 35155  df-atl 35186  df-cvlat 35210  df-hlat 35239  df-llines 35386  df-lplanes 35387
This theorem is referenced by:  dalemdnee  35554
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