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Theorem dlatl 17803
Description: A distributive lattice is a lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
dlatl (𝐾 ∈ DLat → 𝐾 ∈ Lat)

Proof of Theorem dlatl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2824 . . 3 (join‘𝐾) = (join‘𝐾)
3 eqid 2824 . . 3 (meet‘𝐾) = (meet‘𝐾)
41, 2, 3isdlat 17801 . 2 (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(meet‘𝐾)(𝑦(join‘𝐾)𝑧)) = ((𝑥(meet‘𝐾)𝑦)(join‘𝐾)(𝑥(meet‘𝐾)𝑧))))
54simplbi 501 1 (𝐾 ∈ DLat → 𝐾 ∈ Lat)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wral 3133  cfv 6344  (class class class)co 7146  Basecbs 16481  joincjn 17552  meetcmee 17553  Latclat 17653  DLatcdlat 17799
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5197
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-iota 6303  df-fv 6352  df-ov 7149  df-dlat 17800
This theorem is referenced by: (None)
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