Theorem atllat 34587

Description: An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)

Assertion

Ref	Expression
atllat	|- ( K e. AtLat -> K e. Lat )

Proof of Theorem atllat

Dummy variables x p 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 |- ( glb ` K ) = ( glb ` K )
3		eqid 2622	. . 3 |- ( le ` K ) = ( le ` K )
4		eqid 2622	. . 3 |- ( 0. ` K ) = ( 0. ` K )
5		eqid 2622	. . 3 |- ( Atoms ` K ) = ( Atoms ` K )
6	1, 2, 3, 4, 5	isatl 34586	. 2 |- ( K e. AtLat <-> ( K e. Lat /\ ( Base ` K ) e. dom ( glb ` K ) /\ A. x e. ( Base ` K ) ( x =/= ( 0. ` K ) -> E. p e. ( Atoms ` K ) p ( le ` K ) x ) ) )
7	6	simp1bi 1076	1 |- ( K e. AtLat -> K e. Lat )

Syntax hints:   -> wi 4   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   class class class wbr 4653   dom cdm 5114   ` cfv 5888   Basecbs 15857   lecple 15948   glbcglb 16943   0.cp0 17037   Latclat 17045   Atomscatm 34550   AtLatcal 34551

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-ne 2795  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-dm 5124  df-iota 5851  df-fv 5896  df-atl 34585

This theorem is referenced by:  atlpos  34588  atnle  34604  atlatmstc  34606  cvllat  34613  hllat  34650  snatpsubN  35036