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Theorem islati 14158
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 518 . . 3  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( ( x  .\/  y )  e.  B  /\  ( x  ./\  y
)  e.  B ) )
54rgen2a 2609 . 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 14153 . 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 886 1  |-  K  e. 
Lat
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   Posetcpo 14074   joincjn 14078   meetcmee 14079   Latclat 14151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-lat 14152
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