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Theorem islati 14444
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 519 . . 3  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( ( x  .\/  y )  e.  B  /\  ( x  ./\  y
)  e.  B ) )
54rgen2a 2740 . 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 14439 . 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 887 1  |-  K  e. 
Lat
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   ` cfv 5421  (class class class)co 6048   Basecbs 13432   Posetcpo 14360   joincjn 14364   meetcmee 14365   Latclat 14437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-ov 6051  df-lat 14438
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