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Theorem isclat 14231
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 5541 . . . . . 6  |-  ( l  =  K  ->  ( Base `  l )  =  ( Base `  K
) )
2 isclat.b . . . . . 6  |-  B  =  ( Base `  K
)
31, 2syl6eqr 2346 . . . . 5  |-  ( l  =  K  ->  ( Base `  l )  =  B )
43sseq2d 3219 . . . 4  |-  ( l  =  K  ->  (
s  C_  ( Base `  l )  <->  s  C_  B ) )
5 fveq2 5541 . . . . . . . 8  |-  ( l  =  K  ->  ( lub `  l )  =  ( lub `  K
) )
6 isclat.u . . . . . . . 8  |-  U  =  ( lub `  K
)
75, 6syl6eqr 2346 . . . . . . 7  |-  ( l  =  K  ->  ( lub `  l )  =  U )
87fveq1d 5543 . . . . . 6  |-  ( l  =  K  ->  (
( lub `  l
) `  s )  =  ( U `  s ) )
98, 3eleq12d 2364 . . . . 5  |-  ( l  =  K  ->  (
( ( lub `  l
) `  s )  e.  ( Base `  l
)  <->  ( U `  s )  e.  B
) )
10 fveq2 5541 . . . . . . . 8  |-  ( l  =  K  ->  ( glb `  l )  =  ( glb `  K
) )
11 isclat.g . . . . . . . 8  |-  G  =  ( glb `  K
)
1210, 11syl6eqr 2346 . . . . . . 7  |-  ( l  =  K  ->  ( glb `  l )  =  G )
1312fveq1d 5543 . . . . . 6  |-  ( l  =  K  ->  (
( glb `  l
) `  s )  =  ( G `  s ) )
1413, 3eleq12d 2364 . . . . 5  |-  ( l  =  K  ->  (
( ( glb `  l
) `  s )  e.  ( Base `  l
)  <->  ( G `  s )  e.  B
) )
159, 14anbi12d 691 . . . 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 311 . . 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 1615 . 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 14230 . 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 2938 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 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271   Basecbs 13164   Posetcpo 14090   lubclub 14092   glbcglb 14093   CLatccla 14229
This theorem is referenced by:  clatlem  14232  isclati  14235  clatl  14236  oduclatb  14264  mreclat  14306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-clat 14230
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