MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dlatl Structured version   Visualization version   GIF version

Theorem dlatl 18569
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 2737 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2737 . . 3 (join‘𝐾) = (join‘𝐾)
3 eqid 2737 . . 3 (meet‘𝐾) = (meet‘𝐾)
41, 2, 3isdlat 18567 . 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 497 1 (𝐾 ∈ DLat → 𝐾 ∈ Lat)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061  cfv 6561  (class class class)co 7431  Basecbs 17247  joincjn 18357  meetcmee 18358  Latclat 18476  DLatcdlat 18565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-dlat 18566
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator