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Theorem dalemkelat 39227
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
dalemkelat (𝜑𝐾 ∈ Lat)

Proof of Theorem dalemkelat
StepHypRef Expression
1 dalema.ph . . 3 (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)))))
21dalemkehl 39226 . 2 (𝜑𝐾 ∈ HL)
32hllatd 38966 1 (𝜑𝐾 ∈ Lat)
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  Latclat 18426  HLchlt 38952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-dm 5688  df-iota 6501  df-fv 6557  df-ov 7422  df-atl 38900  df-cvlat 38924  df-hlat 38953
This theorem is referenced by:  dalemcnes  39253  dalempnes  39254  dalemqnet  39255  dalemply  39257  dalemsly  39258  dalem1  39262  dalemcea  39263  dalem3  39267  dalem4  39268  dalem5  39270  dalem8  39273  dalem-cly  39274  dalem10  39276  dalem13  39279  dalem16  39282  dalem17  39283  dalem21  39297  dalem25  39301  dalem27  39302  dalem38  39313  dalem39  39314  dalem43  39318  dalem44  39319  dalem45  39320  dalem48  39323  dalem54  39329  dalem55  39330  dalem56  39331  dalem57  39332  dalem60  39335
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