Theorem dlatl 18459

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 2737	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2737	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2737	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
4	1, 2, 3	isdlat 18457	. 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 496	1 ⊢ (𝐾 ∈ DLat → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  joincjn 18246  meetcmee 18247  Latclat 18366  DLatcdlat 18455

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-dlat 18456

This theorem is referenced by: (None)