Theorem dlatl 17195

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 2622	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2622	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2622	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
4	1, 2, 3	isdlat 17193	. 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 476	1 ⊢ (𝐾 ∈ DLat → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  joincjn 16944  meetcmee 16945  Latclat 17045  DLatcdlat 17191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-dlat 17192

This theorem is referenced by: (None)