Theorem cvlatl 39824

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 2740	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2740	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2740	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2740	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	1, 2, 3, 4	iscvlat 39822	. 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 396   ∈ wcel 2119  ∀wral 3054   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  lecple 17225  joincjn 18275  Atomscatm 39762  AtLatcal 39763  CvLatclc 39764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-cvlat 39821

This theorem is referenced by:  cvllat  39825  cvlexch3  39831  cvlexch4N  39832  cvlatexchb1  39833  cvlcvr1  39838  cvlcvrp  39839  cvlatcvr1  39840  cvlsupr2  39842  hlatl  39859