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Definition df-cvlat 30057
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 30000 . 2  class  CvLat
2 va . . . . . . . . . . 11  set  a
32cv 1651 . . . . . . . . . 10  class  a
4 vc . . . . . . . . . . 11  set  c
54cv 1651 . . . . . . . . . 10  class  c
6 vk . . . . . . . . . . . 12  set  k
76cv 1651 . . . . . . . . . . 11  class  k
8 cple 13528 . . . . . . . . . . 11  class  le
97, 8cfv 5446 . . . . . . . . . 10  class  ( le
`  k )
103, 5, 9wbr 4204 . . . . . . . . 9  wff  a ( le `  k ) c
1110wn 3 . . . . . . . 8  wff  -.  a
( le `  k
) c
12 vb . . . . . . . . . . 11  set  b
1312cv 1651 . . . . . . . . . 10  class  b
14 cjn 14393 . . . . . . . . . . 11  class  join
157, 14cfv 5446 . . . . . . . . . 10  class  ( join `  k )
165, 13, 15co 6073 . . . . . . . . 9  class  ( c ( join `  k
) b )
173, 16, 9wbr 4204 . . . . . . . 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 6073 . . . . . . . 8  class  ( c ( join `  k
) a )
2013, 19, 9wbr 4204 . . . . . . 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 13461 . . . . . . 7  class  Base
237, 22cfv 5446 . . . . . 6  class  ( Base `  k )
2421, 4, 23wral 2697 . . . . 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 29998 . . . . . 6  class  Atoms
267, 25cfv 5446 . . . . 5  class  ( Atoms `  k )
2724, 12, 26wral 2697 . . . 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 2697 . . 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 29999 . . 3  class  AtLat
3028, 6, 29crab 2701 . 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 1652 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  30058
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