Theorem cvlatl 39325

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 2730	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2730	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2730	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2730	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	1, 2, 3, 4	iscvlat 39323	. 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 497	1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3045   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234  joincjn 18279  Atomscatm 39263  AtLatcal 39264  CvLatclc 39265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393  df-cvlat 39322

This theorem is referenced by:  cvllat  39326  cvlexch3  39332  cvlexch4N  39333  cvlatexchb1  39334  cvlcvr1  39339  cvlcvrp  39340  cvlatcvr1  39341  cvlsupr2  39343  hlatl  39360