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Theorem dalemkehl 39625
Description: Lemma for dath 39738. 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 1285 . 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 217 1 (𝜑𝐾 ∈ HL)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087  wcel 2108   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  HLchlt 39351
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 207  df-an 396  df-3an 1089
This theorem is referenced by:  dalemkelat  39626  dalemkeop  39627  dalempjqeb  39647  dalemsjteb  39648  dalemtjueb  39649  dalemqrprot  39650  dalempnes  39653  dalemqnet  39654  dalempjsen  39655  dalemply  39656  dalemsly  39657  dalemswapyz  39658  dalemrot  39659  dalemrotyz  39660  dalem1  39661  dalemcea  39662  dalem2  39663  dalemdea  39664  dalem3  39666  dalem4  39667  dalem5  39669  dalem-cly  39673  dalem9  39674  dalem11  39676  dalem12  39677  dalem13  39678  dalem15  39680  dalem16  39681  dalem17  39682  dalem18  39683  dalem19  39684  dalemswapyzps  39692  dalemcjden  39694  dalem21  39696  dalem22  39697  dalem23  39698  dalem24  39699  dalem25  39700  dalem27  39701  dalem28  39702  dalem38  39712  dalem39  39713  dalem41  39715  dalem42  39716  dalem43  39717  dalem44  39718  dalem45  39719  dalem51  39725  dalem52  39726  dalem54  39728  dalem55  39729  dalem56  39730  dalem57  39731  dalem58  39732  dalem59  39733  dalem60  39734
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