Theorem dlatl 17018

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.

Step	Hyp	Ref	Expression
1		eqid 2610	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2610	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2610	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
4	1, 2, 3	isdlat 17016	. 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))))
5	4	simplbi 475	1 ⊢ (𝐾 ∈ DLat → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  joincjn 16767  meetcmee 16768  Latclat 16868  DLatcdlat 17014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-dlat 17015

This theorem is referenced by: (None)