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Theorem islat 15217
Description: The predicate "is a lattice." (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
islat.b  |-  B  =  ( Base `  K
)
islat.j  |-  .\/  =  ( join `  K )
islat.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
islat  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  ( dom  .\/  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) ) )

Proof of Theorem islat
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 fveq2 5691 . . . . . 6  |-  ( l  =  K  ->  ( join `  l )  =  ( join `  K
) )
2 islat.j . . . . . 6  |-  .\/  =  ( join `  K )
31, 2syl6eqr 2493 . . . . 5  |-  ( l  =  K  ->  ( join `  l )  = 
.\/  )
43dmeqd 5042 . . . 4  |-  ( l  =  K  ->  dom  ( join `  l )  =  dom  .\/  )
5 fveq2 5691 . . . . . 6  |-  ( l  =  K  ->  ( Base `  l )  =  ( Base `  K
) )
6 islat.b . . . . . 6  |-  B  =  ( Base `  K
)
75, 6syl6eqr 2493 . . . . 5  |-  ( l  =  K  ->  ( Base `  l )  =  B )
87, 7xpeq12d 4865 . . . 4  |-  ( l  =  K  ->  (
( Base `  l )  X.  ( Base `  l
) )  =  ( B  X.  B ) )
94, 8eqeq12d 2457 . . 3  |-  ( l  =  K  ->  ( dom  ( join `  l
)  =  ( (
Base `  l )  X.  ( Base `  l
) )  <->  dom  .\/  =  ( B  X.  B
) ) )
10 fveq2 5691 . . . . . 6  |-  ( l  =  K  ->  ( meet `  l )  =  ( meet `  K
) )
11 islat.m . . . . . 6  |-  ./\  =  ( meet `  K )
1210, 11syl6eqr 2493 . . . . 5  |-  ( l  =  K  ->  ( meet `  l )  = 
./\  )
1312dmeqd 5042 . . . 4  |-  ( l  =  K  ->  dom  ( meet `  l )  =  dom  ./\  )
1413, 8eqeq12d 2457 . . 3  |-  ( l  =  K  ->  ( dom  ( meet `  l
)  =  ( (
Base `  l )  X.  ( Base `  l
) )  <->  dom  ./\  =  ( B  X.  B
) ) )
159, 14anbi12d 710 . 2  |-  ( l  =  K  ->  (
( dom  ( join `  l )  =  ( ( Base `  l
)  X.  ( Base `  l ) )  /\  dom  ( meet `  l
)  =  ( (
Base `  l )  X.  ( Base `  l
) ) )  <->  ( dom  .\/  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) ) )
16 df-lat 15216 . 2  |-  Lat  =  { l  e.  Poset  |  ( dom  ( join `  l )  =  ( ( Base `  l
)  X.  ( Base `  l ) )  /\  dom  ( meet `  l
)  =  ( (
Base `  l )  X.  ( Base `  l
) ) ) }
1715, 16elrab2 3119 1  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  ( dom  .\/  =  ( B  X.  B )  /\  dom  ./\  =  ( B  X.  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    X. cxp 4838   dom cdm 4840   ` cfv 5418   Basecbs 14174   Posetcpo 15110   joincjn 15114   meetcmee 15115   Latclat 15215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-xp 4846  df-dm 4850  df-iota 5381  df-fv 5426  df-lat 15216
This theorem is referenced by:  latcl2  15218  latlem  15219  latpos  15220  latjcom  15229  latmcom  15245  clatl  15286  odulatb  15313
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