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Theorem dalemkehl 39226
Description: Lemma for dath 39339. 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 1281 . 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 216 1 (𝜑𝐾 ∈ HL)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084  wcel 2098   class class class wbr 5149  cfv 6549  (class class class)co 7419  Basecbs 17183  HLchlt 38952
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 206  df-an 395  df-3an 1086
This theorem is referenced by:  dalemkelat  39227  dalemkeop  39228  dalempjqeb  39248  dalemsjteb  39249  dalemtjueb  39250  dalemqrprot  39251  dalempnes  39254  dalemqnet  39255  dalempjsen  39256  dalemply  39257  dalemsly  39258  dalemswapyz  39259  dalemrot  39260  dalemrotyz  39261  dalem1  39262  dalemcea  39263  dalem2  39264  dalemdea  39265  dalem3  39267  dalem4  39268  dalem5  39270  dalem-cly  39274  dalem9  39275  dalem11  39277  dalem12  39278  dalem13  39279  dalem15  39281  dalem16  39282  dalem17  39283  dalem18  39284  dalem19  39285  dalemswapyzps  39293  dalemcjden  39295  dalem21  39297  dalem22  39298  dalem23  39299  dalem24  39300  dalem25  39301  dalem27  39302  dalem28  39303  dalem38  39313  dalem39  39314  dalem41  39316  dalem42  39317  dalem43  39318  dalem44  39319  dalem45  39320  dalem51  39326  dalem52  39327  dalem54  39329  dalem55  39330  dalem56  39331  dalem57  39332  dalem58  39333  dalem59  39334  dalem60  39335
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