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Definition df-cvlat 29805
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 29748 . 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 13491 . . . . . . . . . . 11  class  le
97, 8cfv 5413 . . . . . . . . . 10  class  ( le
`  k )
103, 5, 9wbr 4172 . . . . . . . . 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 14356 . . . . . . . . . . 11  class  join
157, 14cfv 5413 . . . . . . . . . 10  class  ( join `  k )
165, 13, 15co 6040 . . . . . . . . 9  class  ( c ( join `  k
) b )
173, 16, 9wbr 4172 . . . . . . . 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 6040 . . . . . . . 8  class  ( c ( join `  k
) a )
2013, 19, 9wbr 4172 . . . . . . 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 13424 . . . . . . 7  class  Base
237, 22cfv 5413 . . . . . 6  class  ( Base `  k )
2421, 4, 23wral 2666 . . . . 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 29746 . . . . . 6  class  Atoms
267, 25cfv 5413 . . . . 5  class  ( Atoms `  k )
2724, 12, 26wral 2666 . . . 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 2666 . . 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 29747 . . 3  class  AtLat
3028, 6, 29crab 2670 . 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  29806
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