MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  islati Unicode version

Theorem islati 14251
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 2685 . 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 14246 . 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 1642    e. wcel 1710   A.wral 2619   ` cfv 5334  (class class class)co 5942   Basecbs 13239   Posetcpo 14167   joincjn 14171   meetcmee 14172   Latclat 14244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-iota 5298  df-fv 5342  df-ov 5945  df-lat 14245
  Copyright terms: Public domain W3C validator