Theorem cvlatl 34612

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 2622	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2622	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2622	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2622	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	1, 2, 3, 4	iscvlat 34610	. 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 476	1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∈ wcel 1990  ∀wral 2912   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  lecple 15948  joincjn 16944  Atomscatm 34550  AtLatcal 34551  CvLatclc 34552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-cvlat 34609

This theorem is referenced by:  cvllat  34613  cvlexch3  34619  cvlexch4N  34620  cvlatexchb1  34621  cvlcvr1  34626  cvlcvrp  34627  cvlatcvr1  34628  cvlsupr2  34630  hlatl  34647