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Metamath Proof Explorer

Theorem isclat

Description: The predicate "is a complete lattice". (Contributed by NM, 18-Oct-2012) (Revised by NM, 12-Sep-2018)

Ref Expression
Hypotheses isclat.b 
|- B = ( Base ` K )
isclat.u 
|- U = ( lub ` K )
isclat.g 
|- G = ( glb ` K )
Assertion isclat 
|- ( K e. CLat <-> ( K e. Poset /\ ( dom U = ~P B /\ dom G = ~P B ) ) )

Proof

Step Hyp Ref Expression
1 isclat.b 
 |-  B = ( Base ` K )
2 isclat.u 
 |-  U = ( lub ` K )
3 isclat.g 
 |-  G = ( glb ` K )
4 fveq2 
 |-  ( l = K -> ( lub ` l ) = ( lub ` K ) )
5 4 2 eqtr4di 
 |-  ( l = K -> ( lub ` l ) = U )
6 5 dmeqd 
 |-  ( l = K -> dom ( lub ` l ) = dom U )
7 fveq2 
 |-  ( l = K -> ( Base ` l ) = ( Base ` K ) )
8 7 1 eqtr4di 
 |-  ( l = K -> ( Base ` l ) = B )
9 8 pweqd 
 |-  ( l = K -> ~P ( Base ` l ) = ~P B )
10 6 9 eqeq12d 
 |-  ( l = K -> ( dom ( lub ` l ) = ~P ( Base ` l ) <-> dom U = ~P B ) )
11 fveq2 
 |-  ( l = K -> ( glb ` l ) = ( glb ` K ) )
12 11 3 eqtr4di 
 |-  ( l = K -> ( glb ` l ) = G )
13 12 dmeqd 
 |-  ( l = K -> dom ( glb ` l ) = dom G )
14 13 9 eqeq12d 
 |-  ( l = K -> ( dom ( glb ` l ) = ~P ( Base ` l ) <-> dom G = ~P B ) )
15 10 14 anbi12d 
 |-  ( l = K -> ( ( dom ( lub ` l ) = ~P ( Base ` l ) /\ dom ( glb ` l ) = ~P ( Base ` l ) ) <-> ( dom U = ~P B /\ dom G = ~P B ) ) )
16 df-clat 
 |-  CLat = { l e. Poset | ( dom ( lub ` l ) = ~P ( Base ` l ) /\ dom ( glb ` l ) = ~P ( Base ` l ) ) }
17 15 16 elrab2 
 |-  ( K e. CLat <-> ( K e. Poset /\ ( dom U = ~P B /\ dom G = ~P B ) ) )