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Definition df-cvlat 29488
Description: Define the class of atomic lattices with the covering property. (This is actually the exchange property, but they are equivalent. The literature usually uses the covering property terminology.) (Contributed by NM, 5-Nov-2012.)
Assertion
Ref Expression
df-cvlat  |-  CvLat  =  {
k  e.  AtLat  |  A. a  e.  ( Atoms `  k ) A. b  e.  ( Atoms `  k ) A. c  e.  ( Base `  k ) ( ( -.  a ( le `  k ) c  /\  a ( le `  k ) ( c ( join `  k ) b ) )  ->  b ( le `  k ) ( c ( join `  k
) a ) ) }
Distinct variable group:    k, c, a, b

Detailed syntax breakdown of Definition df-cvlat
StepHypRef Expression
1 clc 29431 . 2  class  CvLat
2 va . . . . . . . . . . 11  set  a
32cv 1648 . . . . . . . . . 10  class  a
4 vc . . . . . . . . . . 11  set  c
54cv 1648 . . . . . . . . . 10  class  c
6 vk . . . . . . . . . . . 12  set  k
76cv 1648 . . . . . . . . . . 11  class  k
8 cple 13456 . . . . . . . . . . 11  class  le
97, 8cfv 5387 . . . . . . . . . 10  class  ( le
`  k )
103, 5, 9wbr 4146 . . . . . . . . 9  wff  a ( le `  k ) c
1110wn 3 . . . . . . . 8  wff  -.  a
( le `  k
) c
12 vb . . . . . . . . . . 11  set  b
1312cv 1648 . . . . . . . . . 10  class  b
14 cjn 14321 . . . . . . . . . . 11  class  join
157, 14cfv 5387 . . . . . . . . . 10  class  ( join `  k )
165, 13, 15co 6013 . . . . . . . . 9  class  ( c ( join `  k
) b )
173, 16, 9wbr 4146 . . . . . . . 8  wff  a ( le `  k ) ( c ( join `  k ) b )
1811, 17wa 359 . . . . . . 7  wff  ( -.  a ( le `  k ) c  /\  a ( le `  k ) ( c ( join `  k
) b ) )
195, 3, 15co 6013 . . . . . . . 8  class  ( c ( join `  k
) a )
2013, 19, 9wbr 4146 . . . . . . 7  wff  b ( le `  k ) ( c ( join `  k ) a )
2118, 20wi 4 . . . . . 6  wff  ( ( -.  a ( le
`  k ) c  /\  a ( le
`  k ) ( c ( join `  k
) b ) )  ->  b ( le
`  k ) ( c ( join `  k
) a ) )
22 cbs 13389 . . . . . . 7  class  Base
237, 22cfv 5387 . . . . . 6  class  ( Base `  k )
2421, 4, 23wral 2642 . . . . 5  wff  A. c  e.  ( Base `  k
) ( ( -.  a ( le `  k ) c  /\  a ( le `  k ) ( c ( join `  k
) b ) )  ->  b ( le
`  k ) ( c ( join `  k
) a ) )
25 catm 29429 . . . . . 6  class  Atoms
267, 25cfv 5387 . . . . 5  class  ( Atoms `  k )
2724, 12, 26wral 2642 . . . 4  wff  A. b  e.  ( Atoms `  k ) A. c  e.  ( Base `  k ) ( ( -.  a ( le `  k ) c  /\  a ( le `  k ) ( c ( join `  k ) b ) )  ->  b ( le `  k ) ( c ( join `  k
) a ) )
2827, 2, 26wral 2642 . . 3  wff  A. a  e.  ( Atoms `  k ) A. b  e.  ( Atoms `  k ) A. c  e.  ( Base `  k ) ( ( -.  a ( le
`  k ) c  /\  a ( le
`  k ) ( c ( join `  k
) b ) )  ->  b ( le
`  k ) ( c ( join `  k
) a ) )
29 cal 29430 . . 3  class  AtLat
3028, 6, 29crab 2646 . 2  class  { k  e.  AtLat  |  A. a  e.  ( Atoms `  k ) A. b  e.  ( Atoms `  k ) A. c  e.  ( Base `  k ) ( ( -.  a ( le
`  k ) c  /\  a ( le
`  k ) ( c ( join `  k
) b ) )  ->  b ( le
`  k ) ( c ( join `  k
) a ) ) }
311, 30wceq 1649 1  wff  CvLat  =  {
k  e.  AtLat  |  A. a  e.  ( Atoms `  k ) A. b  e.  ( Atoms `  k ) A. c  e.  ( Base `  k ) ( ( -.  a ( le `  k ) c  /\  a ( le `  k ) ( c ( join `  k ) b ) )  ->  b ( le `  k ) ( c ( join `  k
) a ) ) }
Colors of variables: wff set class
This definition is referenced by:  iscvlat  29489
  Copyright terms: Public domain W3C validator