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 e. CvLat -> K e. 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 e. CvLat <-> ( K e. AtLat /\ A. p e. ( Atoms ` K ) A. q e. ( Atoms ` K ) A. x e. ( 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 e. CvLat -> K e. AtLat )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   e. wcel 1990   A.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