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Theorem isclat 14467
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 5670 . . . . . 6  |-  ( l  =  K  ->  ( Base `  l )  =  ( Base `  K
) )
2 isclat.b . . . . . 6  |-  B  =  ( Base `  K
)
31, 2syl6eqr 2439 . . . . 5  |-  ( l  =  K  ->  ( Base `  l )  =  B )
43sseq2d 3321 . . . 4  |-  ( l  =  K  ->  (
s  C_  ( Base `  l )  <->  s  C_  B ) )
5 fveq2 5670 . . . . . . . 8  |-  ( l  =  K  ->  ( lub `  l )  =  ( lub `  K
) )
6 isclat.u . . . . . . . 8  |-  U  =  ( lub `  K
)
75, 6syl6eqr 2439 . . . . . . 7  |-  ( l  =  K  ->  ( lub `  l )  =  U )
87fveq1d 5672 . . . . . 6  |-  ( l  =  K  ->  (
( lub `  l
) `  s )  =  ( U `  s ) )
98, 3eleq12d 2457 . . . . 5  |-  ( l  =  K  ->  (
( ( lub `  l
) `  s )  e.  ( Base `  l
)  <->  ( U `  s )  e.  B
) )
10 fveq2 5670 . . . . . . . 8  |-  ( l  =  K  ->  ( glb `  l )  =  ( glb `  K
) )
11 isclat.g . . . . . . . 8  |-  G  =  ( glb `  K
)
1210, 11syl6eqr 2439 . . . . . . 7  |-  ( l  =  K  ->  ( glb `  l )  =  G )
1312fveq1d 5672 . . . . . 6  |-  ( l  =  K  ->  (
( glb `  l
) `  s )  =  ( G `  s ) )
1413, 3eleq12d 2457 . . . . 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 1632 . 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 14466 . 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 3039 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 1546    = wceq 1649    e. wcel 1717    C_ wss 3265   ` cfv 5396   Basecbs 13398   Posetcpo 14326   lubclub 14328   glbcglb 14329   CLatccla 14465
This theorem is referenced by:  clatlem  14468  isclati  14471  clatl  14472  oduclatb  14500  mreclat  14542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-iota 5360  df-fv 5404  df-clat 14466
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