Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem4 Structured version   Visualization version   GIF version

Theorem dalem4 33768
Description: Lemma for dalemdnee 33769. (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 33759 . . . 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 479 . . 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 33727 . . . . . . 7 (𝜑𝐾 ∈ Lat)
91, 3, 4dalempjqeb 33748 . . . . . . 7 (𝜑 → (𝑃 𝑄) ∈ (Base‘𝐾))
101, 3, 4dalemsjteb 33749 . . . . . . 7 (𝜑 → (𝑆 𝑇) ∈ (Base‘𝐾))
11 eqid 2605 . . . . . . . 8 (Base‘𝐾) = (Base‘𝐾)
12 dalem3.m . . . . . . . 8 = (meet‘𝐾)
1311, 12latmcom 16840 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑆 𝑇) (𝑃 𝑄)))
148, 9, 10, 13syl3anc 1317 . . . . . 6 (𝜑 → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑆 𝑇) (𝑃 𝑄)))
157, 14syl5eq 2651 . . . . 5 (𝜑𝐷 = ((𝑆 𝑇) (𝑃 𝑄)))
1615neeq1d 2836 . . . 4 (𝜑 → (𝐷𝑇 ↔ ((𝑆 𝑇) (𝑃 𝑄)) ≠ 𝑇))
1716biimpa 499 . . 3 ((𝜑𝐷𝑇) → ((𝑆 𝑇) (𝑃 𝑄)) ≠ 𝑇)
18 biid 249 . . . 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 2605 . . . 4 ((𝑆 𝑇) (𝑃 𝑄)) = ((𝑆 𝑇) (𝑃 𝑄))
23 eqid 2605 . . . 4 ((𝑇 𝑈) (𝑄 𝑅)) = ((𝑇 𝑈) (𝑄 𝑅))
2418, 2, 3, 4, 12, 19, 20, 21, 22, 23dalem3 33767 . . 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 690 . 2 ((𝜑𝐷𝑇) → ((𝑆 𝑇) (𝑃 𝑄)) ≠ ((𝑇 𝑈) (𝑄 𝑅)))
2615adantr 479 . 2 ((𝜑𝐷𝑇) → 𝐷 = ((𝑆 𝑇) (𝑃 𝑄)))
27 dalem3.e . . . 4 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
281dalemkehl 33726 . . . . . 6 (𝜑𝐾 ∈ HL)
291dalemqea 33730 . . . . . 6 (𝜑𝑄𝐴)
301dalemrea 33731 . . . . . 6 (𝜑𝑅𝐴)
3111, 3, 4hlatjcl 33470 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
3228, 29, 30, 31syl3anc 1317 . . . . 5 (𝜑 → (𝑄 𝑅) ∈ (Base‘𝐾))
331, 3, 4dalemtjueb 33750 . . . . 5 (𝜑 → (𝑇 𝑈) ∈ (Base‘𝐾))
3411, 12latmcom 16840 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ (𝑇 𝑈) ∈ (Base‘𝐾)) → ((𝑄 𝑅) (𝑇 𝑈)) = ((𝑇 𝑈) (𝑄 𝑅)))
358, 32, 33, 34syl3anc 1317 . . . 4 (𝜑 → ((𝑄 𝑅) (𝑇 𝑈)) = ((𝑇 𝑈) (𝑄 𝑅)))
3627, 35syl5eq 2651 . . 3 (𝜑𝐸 = ((𝑇 𝑈) (𝑄 𝑅)))
3736adantr 479 . 2 ((𝜑𝐷𝑇) → 𝐸 = ((𝑇 𝑈) (𝑄 𝑅)))
3825, 26, 373netr4d 2854 1 ((𝜑𝐷𝑇) → 𝐷𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775   class class class wbr 4573  cfv 5786  (class class class)co 6523  Basecbs 15637  lecple 15717  joincjn 16709  meetcmee 16710  Latclat 16810  Atomscatm 33367  HLchlt 33454  LPlanesclpl 33595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-preset 16693  df-poset 16711  df-plt 16723  df-lub 16739  df-glb 16740  df-join 16741  df-meet 16742  df-p0 16804  df-lat 16811  df-clat 16873  df-oposet 33280  df-ol 33282  df-oml 33283  df-covers 33370  df-ats 33371  df-atl 33402  df-cvlat 33426  df-hlat 33455  df-llines 33601  df-lplanes 33602
This theorem is referenced by:  dalemdnee  33769
  Copyright terms: Public domain W3C validator