Theorem dlatl 18584

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 2763	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2763	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2763	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
4	1, 2, 3	isdlat 18582	. 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 501	1 ⊢ (𝐾 ∈ DLat → 𝐾 ∈ Lat)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ‘cfv 6536  (class class class)co 7410  Basecbs 17273  joincjn 18371  meetcmee 18372  Latclat 18491  DLatcdlat 18580

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-dlat 18581

This theorem is used by: (None)