Theorem cvlatl 39984

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

Assertion

Ref	Expression
cvlatl	⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)

Proof of Theorem cvlatl

Dummy variables 𝑞 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2769	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2769	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2769	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2769	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	1, 2, 3, 4	iscvlat 39982	. 2 ⊢ (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)∀𝑥 ∈ (Base‘𝐾)((¬ 𝑝(le‘𝐾)𝑥 ∧ 𝑝(le‘𝐾)(𝑥(join‘𝐾)𝑞)) → 𝑞(le‘𝐾)(𝑥(join‘𝐾)𝑝))))
6	5	simplbi 501	1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∈ wcel 2149  ∀wral 3085   class class class wbr 5110  ‘cfv 6533  (class class class)co 7408  Basecbs 17265  lecple 17313  joincjn 18363  Atomscatm 39922  AtLatcal 39923  CvLatclc 39924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-iota 6489  df-fv 6541  df-ov 7411  df-cvlat 39981

This theorem is referenced by:  cvllat  39985  cvlexch3  39991  cvlexch4N  39992  cvlatexchb1  39993  cvlcvr1  39998  cvlcvrp  39999  cvlatcvr1  40000  cvlsupr2  40002  hlatl  40019