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Theorem dalemqnet 33759
Description: Lemma for dath 33843. 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 33731 . . 3 (𝜑𝐾 ∈ Lat)
3 dalemc.a . . . 4 𝐴 = (Atoms‘𝐾)
41, 3dalemceb 33745 . . 3 (𝜑𝐶 ∈ (Base‘𝐾))
51, 3dalemteb 33750 . . 3 (𝜑𝑇 ∈ (Base‘𝐾))
61, 3dalemueb 33751 . . 3 (𝜑𝑈 ∈ (Base‘𝐾))
7 simp322 1204 . . . 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 205 . . 3 (𝜑 → ¬ 𝐶 (𝑇 𝑈))
9 eqid 2609 . . . 4 (Base‘𝐾) = (Base‘𝐾)
10 dalemc.l . . . 4 = (le‘𝐾)
11 dalemc.j . . . 4 = (join‘𝐾)
129, 10, 11latnlej2l 16841 . . 3 ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) ∧ ¬ 𝐶 (𝑇 𝑈)) → ¬ 𝐶 𝑇)
132, 4, 5, 6, 8, 12syl131anc 1330 . 2 (𝜑 → ¬ 𝐶 𝑇)
141dalemclqjt 33742 . . . . 5 (𝜑𝐶 (𝑄 𝑇))
15 oveq1 6534 . . . . . 6 (𝑄 = 𝑇 → (𝑄 𝑇) = (𝑇 𝑇))
1615breq2d 4589 . . . . 5 (𝑄 = 𝑇 → (𝐶 (𝑄 𝑇) ↔ 𝐶 (𝑇 𝑇)))
1714, 16syl5ibcom 233 . . . 4 (𝜑 → (𝑄 = 𝑇𝐶 (𝑇 𝑇)))
181dalemkehl 33730 . . . . . 6 (𝜑𝐾 ∈ HL)
191dalemtea 33737 . . . . . 6 (𝜑𝑇𝐴)
2011, 3hlatjidm 33476 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑇𝐴) → (𝑇 𝑇) = 𝑇)
2118, 19, 20syl2anc 690 . . . . 5 (𝜑 → (𝑇 𝑇) = 𝑇)
2221breq2d 4589 . . . 4 (𝜑 → (𝐶 (𝑇 𝑇) ↔ 𝐶 𝑇))
2317, 22sylibd 227 . . 3 (𝜑 → (𝑄 = 𝑇𝐶 𝑇))
2423necon3bd 2795 . 2 (𝜑 → (¬ 𝐶 𝑇𝑄𝑇))
2513, 24mpd 15 1 (𝜑𝑄𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  Latclat 16814  Atomscatm 33371  HLchlt 33458  LPlanesclpl 33599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
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 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-preset 16697  df-poset 16715  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-lat 16815  df-ats 33375  df-atl 33406  df-cvlat 33430  df-hlat 33459
This theorem is referenced by:  dalemcea  33767  dalem2  33768  dalemdnee  33773
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