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Theorem islati 14486
Description: Properties that determine a lattice. (Contributed by NM, 12-Sep-2011.)
Hypotheses
Ref Expression
islati.1  |-  K  e. 
Poset
islati.b  |-  B  =  ( Base `  K
)
islati.j  |-  .\/  =  ( join `  K )
islati.m  |-  ./\  =  ( meet `  K )
islati.5  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .\/  y
)  e.  B )
islati.6  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  ./\  y
)  e.  B )
Assertion
Ref Expression
islati  |-  K  e. 
Lat
Distinct variable groups:    x, y, B    x, K, y
Allowed substitution hints:    .\/ ( x, y)    ./\ (
x, y)

Proof of Theorem islati
StepHypRef Expression
1 islati.1 . 2  |-  K  e. 
Poset
2 islati.5 . . . 4  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .\/  y
)  e.  B )
3 islati.6 . . . 4  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  ./\  y
)  e.  B )
42, 3jca 520 . . 3  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( ( x  .\/  y )  e.  B  /\  ( x  ./\  y
)  e.  B ) )
54rgen2a 2774 . 2  |-  A. x  e.  B  A. y  e.  B  ( (
x  .\/  y )  e.  B  /\  (
x  ./\  y )  e.  B )
6 islati.b . . 3  |-  B  =  ( Base `  K
)
7 islati.j . . 3  |-  .\/  =  ( join `  K )
8 islati.m . . 3  |-  ./\  =  ( meet `  K )
96, 7, 8islat 14481 . 2  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  A. x  e.  B  A. y  e.  B  ( (
x  .\/  y )  e.  B  /\  (
x  ./\  y )  e.  B ) ) )
101, 5, 9mpbir2an 888 1  |-  K  e. 
Lat
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   ` cfv 5457  (class class class)co 6084   Basecbs 13474   Posetcpo 14402   joincjn 14406   meetcmee 14407   Latclat 14479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087  df-lat 14480
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