MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isclati Structured version   Unicode version

Theorem isclati 14547
Description: Properties that determine a complete lattice. (Contributed by NM, 12-Sep-2011.)
Hypotheses
Ref Expression
isclati.1  |-  K  e. 
Poset
isclati.b  |-  B  =  ( Base `  K
)
isclati.u  |-  U  =  ( lub `  K
)
isclati.g  |-  G  =  ( glb `  K
)
isclati.5  |-  ( s 
C_  B  ->  ( U `  s )  e.  B )
isclati.6  |-  ( s 
C_  B  ->  ( G `  s )  e.  B )
Assertion
Ref Expression
isclati  |-  K  e. 
CLat
Distinct variable group:    K, s
Allowed substitution hints:    B( s)    U( s)    G( s)

Proof of Theorem isclati
StepHypRef Expression
1 isclati.1 . 2  |-  K  e. 
Poset
2 isclati.5 . . . 4  |-  ( s 
C_  B  ->  ( U `  s )  e.  B )
3 isclati.6 . . . 4  |-  ( s 
C_  B  ->  ( G `  s )  e.  B )
42, 3jca 520 . . 3  |-  ( s 
C_  B  ->  (
( U `  s
)  e.  B  /\  ( G `  s )  e.  B ) )
54ax-gen 1556 . 2  |-  A. s
( s  C_  B  ->  ( ( U `  s )  e.  B  /\  ( G `  s
)  e.  B ) )
6 isclati.b . . 3  |-  B  =  ( Base `  K
)
7 isclati.u . . 3  |-  U  =  ( lub `  K
)
8 isclati.g . . 3  |-  G  =  ( glb `  K
)
96, 7, 8isclat 14543 . 2  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  A. s ( s  C_  B  ->  ( ( U `  s
)  e.  B  /\  ( G `  s )  e.  B ) ) ) )
101, 5, 9mpbir2an 888 1  |-  K  e. 
CLat
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726    C_ wss 3322   ` cfv 5457   Basecbs 13474   Posetcpo 14402   lubclub 14404   glbcglb 14405   CLatccla 14541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-clat 14542
  Copyright terms: Public domain W3C validator