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Theorem dalemkelat 36920
Description: Lemma for dath 37032. 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 36919 . 2 (𝜑𝐾 ∈ HL)
32hllatd 36660 1 (𝜑𝐾 ∈ Lat)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084  wcel 2111   class class class wbr 5030  cfv 6324  (class class class)co 7135  Basecbs 16475  Latclat 17647  HLchlt 36646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-dm 5529  df-iota 6283  df-fv 6332  df-ov 7138  df-atl 36594  df-cvlat 36618  df-hlat 36647
This theorem is referenced by:  dalemcnes  36946  dalempnes  36947  dalemqnet  36948  dalemply  36950  dalemsly  36951  dalem1  36955  dalemcea  36956  dalem3  36960  dalem4  36961  dalem5  36963  dalem8  36966  dalem-cly  36967  dalem10  36969  dalem13  36972  dalem16  36975  dalem17  36976  dalem21  36990  dalem25  36994  dalem27  36995  dalem38  37006  dalem39  37007  dalem43  37011  dalem44  37012  dalem45  37013  dalem48  37016  dalem54  37022  dalem55  37023  dalem56  37024  dalem57  37025  dalem60  37028
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