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

Theorem dalemqnet 36941
Description: Lemma for dath 37025. Frequently-used utility lemma. (Contributed by NM, 13-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)
dalempnes.o 𝑂 = (LPlanes‘𝐾)
dalempnes.y 𝑌 = ((𝑃 𝑄) 𝑅)
Assertion
Ref Expression
dalemqnet (𝜑𝑄𝑇)

Proof of Theorem dalemqnet
StepHypRef Expression
1 dalema.ph . . . 4 (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 36913 . . 3 (𝜑𝐾 ∈ Lat)
3 dalemc.a . . . 4 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 36927 . . 3 (𝜑𝐶 ∈ (Base‘𝐾))
51, 3dalemteb 36932 . . 3 (𝜑𝑇 ∈ (Base‘𝐾))
61, 3dalemueb 36933 . . 3 (𝜑𝑈 ∈ (Base‘𝐾))
7 simp322 1321 . . . 4 ((((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)))) → ¬ 𝐶 (𝑇 𝑈))
81, 7sylbi 220 . . 3 (𝜑 → ¬ 𝐶 (𝑇 𝑈))
9 eqid 2801 . . . 4 (Base‘𝐾) = (Base‘𝐾)
10 dalemc.l . . . 4 = (le‘𝐾)
11 dalemc.j . . . 4 = (join‘𝐾)
129, 10, 11latnlej2l 17677 . . 3 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) ∧ ¬ 𝐶 (𝑇 𝑈)) → ¬ 𝐶 𝑇)
132, 4, 5, 6, 8, 12syl131anc 1380 . 2 (𝜑 → ¬ 𝐶 𝑇)
141dalemclqjt 36924 . . . . 5 (𝜑𝐶 (𝑄 𝑇))
15 oveq1 7146 . . . . . 6 (𝑄 = 𝑇 → (𝑄 𝑇) = (𝑇 𝑇))
1615breq2d 5045 . . . . 5 (𝑄 = 𝑇 → (𝐶 (𝑄 𝑇) ↔ 𝐶 (𝑇 𝑇)))
1714, 16syl5ibcom 248 . . . 4 (𝜑 → (𝑄 = 𝑇𝐶 (𝑇 𝑇)))
181dalemkehl 36912 . . . . . 6 (𝜑𝐾 ∈ HL)
191dalemtea 36919 . . . . . 6 (𝜑𝑇𝐴)
2011, 3hlatjidm 36658 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑇𝐴) → (𝑇 𝑇) = 𝑇)
2118, 19, 20syl2anc 587 . . . . 5 (𝜑 → (𝑇 𝑇) = 𝑇)
2221breq2d 5045 . . . 4 (𝜑 → (𝐶 (𝑇 𝑇) ↔ 𝐶 𝑇))
2317, 22sylibd 242 . . 3 (𝜑 → (𝑄 = 𝑇𝐶 𝑇))
2423necon3bd 3004 . 2 (𝜑 → (¬ 𝐶 𝑇𝑄𝑇))
2513, 24mpd 15 1 (𝜑𝑄𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990   class class class wbr 5033  cfv 6328  (class class class)co 7139  Basecbs 16478  lecple 16567  joincjn 17549  Latclat 17650  Atomscatm 36552  HLchlt 36639  LPlanesclpl 36781
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-proset 17533  df-poset 17551  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-lat 17651  df-ats 36556  df-atl 36587  df-cvlat 36611  df-hlat 36640
This theorem is referenced by:  dalemcea  36949  dalem2  36950  dalemdnee  36955
  Copyright terms: Public domain W3C validator