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Theorem islat 15551
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 5872 . . . . . 6  |-  ( l  =  K  ->  ( join `  l )  =  ( join `  K
) )
2 islat.j . . . . . 6  |-  .\/  =  ( join `  K )
31, 2syl6eqr 2526 . . . . 5  |-  ( l  =  K  ->  ( join `  l )  = 
.\/  )
43dmeqd 5211 . . . 4  |-  ( l  =  K  ->  dom  ( join `  l )  =  dom  .\/  )
5 fveq2 5872 . . . . . 6  |-  ( l  =  K  ->  ( Base `  l )  =  ( Base `  K
) )
6 islat.b . . . . . 6  |-  B  =  ( Base `  K
)
75, 6syl6eqr 2526 . . . . 5  |-  ( l  =  K  ->  ( Base `  l )  =  B )
87sqxpeqd 5031 . . . 4  |-  ( l  =  K  ->  (
( Base `  l )  X.  ( Base `  l
) )  =  ( B  X.  B ) )
94, 8eqeq12d 2489 . . 3  |-  ( l  =  K  ->  ( dom  ( join `  l
)  =  ( (
Base `  l )  X.  ( Base `  l
) )  <->  dom  .\/  =  ( B  X.  B
) ) )
10 fveq2 5872 . . . . . 6  |-  ( l  =  K  ->  ( meet `  l )  =  ( meet `  K
) )
11 islat.m . . . . . 6  |-  ./\  =  ( meet `  K )
1210, 11syl6eqr 2526 . . . . 5  |-  ( l  =  K  ->  ( meet `  l )  = 
./\  )
1312dmeqd 5211 . . . 4  |-  ( l  =  K  ->  dom  ( meet `  l )  =  dom  ./\  )
1413, 8eqeq12d 2489 . . 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 15550 . 2  |-  Lat  =  { l  e.  Poset  |  ( dom  ( join `  l )  =  ( ( Base `  l
)  X.  ( Base `  l ) )  /\  dom  ( meet `  l
)  =  ( (
Base `  l )  X.  ( Base `  l
) ) ) }
1715, 16elrab2 3268 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 1379    e. wcel 1767    X. cxp 5003   dom cdm 5005   ` cfv 5594   Basecbs 14507   Posetcpo 15444   joincjn 15448   meetcmee 15449   Latclat 15549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-dm 5015  df-iota 5557  df-fv 5602  df-lat 15550
This theorem is referenced by:  latcl2  15552  latlem  15553  latpos  15554  latjcom  15563  latmcom  15579  clatl  15620  odulatb  15647
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