Theorem dlatl 18582

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 2735	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2735	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2735	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
4	1, 2, 3	isdlat 18580	. 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 497	1 ⊢ (𝐾 ∈ DLat → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  joincjn 18369  meetcmee 18370  Latclat 18489  DLatcdlat 18578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-dlat 18579

This theorem is referenced by: (None)