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Theorem isclat 14530
Description: The predicate "is a complete lattice." (Contributed by NM, 18-Oct-2012.)
Hypotheses
Ref Expression
isclat.b  |-  B  =  ( Base `  K
)
isclat.u  |-  U  =  ( lub `  K
)
isclat.g  |-  G  =  ( glb `  K
)
Assertion
Ref Expression
isclat  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  A. s ( s  C_  B  ->  ( ( U `  s
)  e.  B  /\  ( G `  s )  e.  B ) ) ) )
Distinct variable group:    K, s
Allowed substitution hints:    B( s)    U( s)    G( s)

Proof of Theorem isclat
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . . . . 6  |-  ( l  =  K  ->  ( Base `  l )  =  ( Base `  K
) )
2 isclat.b . . . . . 6  |-  B  =  ( Base `  K
)
31, 2syl6eqr 2485 . . . . 5  |-  ( l  =  K  ->  ( Base `  l )  =  B )
43sseq2d 3368 . . . 4  |-  ( l  =  K  ->  (
s  C_  ( Base `  l )  <->  s  C_  B ) )
5 fveq2 5720 . . . . . . . 8  |-  ( l  =  K  ->  ( lub `  l )  =  ( lub `  K
) )
6 isclat.u . . . . . . . 8  |-  U  =  ( lub `  K
)
75, 6syl6eqr 2485 . . . . . . 7  |-  ( l  =  K  ->  ( lub `  l )  =  U )
87fveq1d 5722 . . . . . 6  |-  ( l  =  K  ->  (
( lub `  l
) `  s )  =  ( U `  s ) )
98, 3eleq12d 2503 . . . . 5  |-  ( l  =  K  ->  (
( ( lub `  l
) `  s )  e.  ( Base `  l
)  <->  ( U `  s )  e.  B
) )
10 fveq2 5720 . . . . . . . 8  |-  ( l  =  K  ->  ( glb `  l )  =  ( glb `  K
) )
11 isclat.g . . . . . . . 8  |-  G  =  ( glb `  K
)
1210, 11syl6eqr 2485 . . . . . . 7  |-  ( l  =  K  ->  ( glb `  l )  =  G )
1312fveq1d 5722 . . . . . 6  |-  ( l  =  K  ->  (
( glb `  l
) `  s )  =  ( G `  s ) )
1413, 3eleq12d 2503 . . . . 5  |-  ( l  =  K  ->  (
( ( glb `  l
) `  s )  e.  ( Base `  l
)  <->  ( G `  s )  e.  B
) )
159, 14anbi12d 692 . . . 4  |-  ( l  =  K  ->  (
( ( ( lub `  l ) `  s
)  e.  ( Base `  l )  /\  (
( glb `  l
) `  s )  e.  ( Base `  l
) )  <->  ( ( U `  s )  e.  B  /\  ( G `  s )  e.  B ) ) )
164, 15imbi12d 312 . . 3  |-  ( l  =  K  ->  (
( s  C_  ( Base `  l )  -> 
( ( ( lub `  l ) `  s
)  e.  ( Base `  l )  /\  (
( glb `  l
) `  s )  e.  ( Base `  l
) ) )  <->  ( s  C_  B  ->  ( ( U `  s )  e.  B  /\  ( G `  s )  e.  B ) ) ) )
1716albidv 1635 . 2  |-  ( l  =  K  ->  ( A. s ( s  C_  ( Base `  l )  ->  ( ( ( lub `  l ) `  s
)  e.  ( Base `  l )  /\  (
( glb `  l
) `  s )  e.  ( Base `  l
) ) )  <->  A. s
( s  C_  B  ->  ( ( U `  s )  e.  B  /\  ( G `  s
)  e.  B ) ) ) )
18 df-clat 14529 . 2  |-  CLat  =  { l  e.  Poset  | 
A. s ( s 
C_  ( Base `  l
)  ->  ( (
( lub `  l
) `  s )  e.  ( Base `  l
)  /\  ( ( glb `  l ) `  s )  e.  (
Base `  l )
) ) }
1917, 18elrab2 3086 1  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  A. s ( s  C_  B  ->  ( ( U `  s
)  e.  B  /\  ( G `  s )  e.  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725    C_ wss 3312   ` cfv 5446   Basecbs 13461   Posetcpo 14389   lubclub 14391   glbcglb 14392   CLatccla 14528
This theorem is referenced by:  clatlem  14531  isclati  14534  clatl  14535  oduclatb  14563  mreclat  14605  xrsclat  24194
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-clat 14529
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