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Metamath Proof Explorer

Theorem iscvlat

Description: The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012)

Ref Expression
Hypotheses iscvlat.b 
|- B = ( Base ` K )
iscvlat.l 
|- .<_ = ( le ` K )
iscvlat.j 
|- .\/ = ( join ` K )
iscvlat.a 
|- A = ( Atoms ` K )
Assertion iscvlat 
|- ( K e. CvLat <-> ( K e. AtLat /\ A. p e. A A. q e. A A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )

Proof

Step Hyp Ref Expression
1 iscvlat.b 
 |-  B = ( Base ` K )
2 iscvlat.l 
 |-  .<_ = ( le ` K )
3 iscvlat.j 
 |-  .\/ = ( join ` K )
4 iscvlat.a 
 |-  A = ( Atoms ` K )
5 fveq2 
 |-  ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) )
6 5 4 eqtr4di 
 |-  ( k = K -> ( Atoms ` k ) = A )
7 fveq2 
 |-  ( k = K -> ( Base ` k ) = ( Base ` K ) )
8 7 1 eqtr4di 
 |-  ( k = K -> ( Base ` k ) = B )
9 fveq2 
 |-  ( k = K -> ( le ` k ) = ( le ` K ) )
10 9 2 eqtr4di 
 |-  ( k = K -> ( le ` k ) = .<_ )
11 10 breqd 
 |-  ( k = K -> ( p ( le ` k ) x <-> p .<_ x ) )
12 11 notbid 
 |-  ( k = K -> ( -. p ( le ` k ) x <-> -. p .<_ x ) )
13 eqidd 
 |-  ( k = K -> p = p )
14 fveq2 
 |-  ( k = K -> ( join ` k ) = ( join ` K ) )
15 14 3 eqtr4di 
 |-  ( k = K -> ( join ` k ) = .\/ )
16 15 oveqd 
 |-  ( k = K -> ( x ( join ` k ) q ) = ( x .\/ q ) )
17 13 10 16 breq123d 
 |-  ( k = K -> ( p ( le ` k ) ( x ( join ` k ) q ) <-> p .<_ ( x .\/ q ) ) )
18 12 17 anbi12d 
 |-  ( k = K -> ( ( -. p ( le ` k ) x /\ p ( le ` k ) ( x ( join ` k ) q ) ) <-> ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) ) )
19 eqidd 
 |-  ( k = K -> q = q )
20 15 oveqd 
 |-  ( k = K -> ( x ( join ` k ) p ) = ( x .\/ p ) )
21 19 10 20 breq123d 
 |-  ( k = K -> ( q ( le ` k ) ( x ( join ` k ) p ) <-> q .<_ ( x .\/ p ) ) )
22 18 21 imbi12d 
 |-  ( k = K -> ( ( ( -. p ( le ` k ) x /\ p ( le ` k ) ( x ( join ` k ) q ) ) -> q ( le ` k ) ( x ( join ` k ) p ) ) <-> ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
23 8 22 raleqbidv 
 |-  ( k = K -> ( A. x e. ( Base ` k ) ( ( -. p ( le ` k ) x /\ p ( le ` k ) ( x ( join ` k ) q ) ) -> q ( le ` k ) ( x ( join ` k ) p ) ) <-> A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
24 6 23 raleqbidv 
 |-  ( k = 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 ) ) <-> A. q e. A A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
25 6 24 raleqbidv 
 |-  ( k = K -> ( 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 ) ) <-> A. p e. A A. q e. A A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
26 df-cvlat 
 |-  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 ) ) }
27 25 26 elrab2 
 |-  ( K e. CvLat <-> ( K e. AtLat /\ A. p e. A A. q e. A A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )