Theorem cvllat 40081

Description: An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)

Assertion

Ref	Expression
cvllat	⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)

Proof of Theorem cvllat

Step	Hyp	Ref	Expression
1		cvlatl 40080	. 2 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
2		atllat 40055	. 2 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
3	1, 2	syl 18	1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)

Syntax hints:   → wi 4   ∈ wcel 2143  Latclat 18488  AtLatcal 40019  CvLatclc 40020

This theorem was proved from 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

This theorem 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-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-dm 5673  df-iota 6494  df-fv 6546  df-ov 7415  df-atl 40053  df-cvlat 40077

This theorem is referenced by:  cvlposN  40082  cvlexch2  40084  cvlexchb1  40085  cvlexchb2  40086  cvlatexchb2  40090  cvlatexch1  40091  cvlatexch2  40092  cvlatexch3  40093  cvlcvr1  40094  cvlcvrp  40095  cvlatcvr2  40097  cvlsupr2  40098  cvlsupr7  40103  cvlsupr8  40104