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Theorem dalemkehl 35424
Description: Lemma for dath 35537. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
Hypothesis
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)))))
Assertion
Ref Expression
dalemkehl (𝜑𝐾 ∈ HL)

Proof of Theorem dalemkehl
StepHypRef Expression
1 dalema.ph . 2 (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 simp11l 1367 . 2 ((((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)))) → 𝐾 ∈ HL)
31, 2sylbi 207 1 (𝜑𝐾 ∈ HL)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1070  wcel 2144   class class class wbr 4784  cfv 6031  (class class class)co 6792  Basecbs 16063  HLchlt 35152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-an 383  df-3an 1072
This theorem is referenced by:  dalemkelat  35425  dalemkeop  35426  dalempjqeb  35446  dalemsjteb  35447  dalemtjueb  35448  dalemqrprot  35449  dalempnes  35452  dalemqnet  35453  dalempjsen  35454  dalemply  35455  dalemsly  35456  dalemswapyz  35457  dalemrot  35458  dalemrotyz  35459  dalem1  35460  dalemcea  35461  dalem2  35462  dalemdea  35463  dalem3  35465  dalem4  35466  dalem5  35468  dalem-cly  35472  dalem9  35473  dalem11  35475  dalem12  35476  dalem13  35477  dalem15  35479  dalem16  35480  dalem17  35481  dalem18  35482  dalem19  35483  dalemswapyzps  35491  dalemcjden  35493  dalem21  35495  dalem22  35496  dalem23  35497  dalem24  35498  dalem25  35499  dalem27  35500  dalem28  35501  dalem38  35511  dalem39  35512  dalem41  35514  dalem42  35515  dalem43  35516  dalem44  35517  dalem45  35518  dalem51  35524  dalem52  35525  dalem54  35527  dalem55  35528  dalem56  35529  dalem57  35530  dalem58  35531  dalem59  35532  dalem60  35533
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