Theorem cvlatl 40198

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 2760	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2760	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2760	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2760	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	1, 2, 3, 4	iscvlat 40196	. 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 502	1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3076   class class class wbr 5103  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  lecple 17349  joincjn 18399  Atomscatm 40136  AtLatcal 40137  CvLatclc 40138

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7416  df-cvlat 40195

This theorem is used by:  cvllat  40199  cvlexch3  40205  cvlexch4N  40206  cvlatexchb1  40207  cvlcvr1  40212  cvlcvrp  40213  cvlatcvr1  40214  cvlsupr2  40216  hlatl  40233