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Theorem latlem 14405
Description: Lemma for lattice properties. (Contributed by NM, 14-Sep-2011.)
Hypotheses
Ref Expression
latlem.b  |-  B  =  ( Base `  K
)
latlem.j  |-  .\/  =  ( join `  K )
latlem.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latlem  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .\/  Y )  e.  B  /\  ( X  ./\  Y )  e.  B ) )

Proof of Theorem latlem
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 latlem.b . . . . 5  |-  B  =  ( Base `  K
)
2 latlem.j . . . . 5  |-  .\/  =  ( join `  K )
3 latlem.m . . . . 5  |-  ./\  =  ( meet `  K )
41, 2, 3islat 14404 . . . 4  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  A. x  e.  B  A. y  e.  B  ( (
x  .\/  y )  e.  B  /\  (
x  ./\  y )  e.  B ) ) )
54simprbi 451 . . 3  |-  ( K  e.  Lat  ->  A. x  e.  B  A. y  e.  B  ( (
x  .\/  y )  e.  B  /\  (
x  ./\  y )  e.  B ) )
6 oveq1 6028 . . . . . 6  |-  ( x  =  X  ->  (
x  .\/  y )  =  ( X  .\/  y ) )
76eleq1d 2454 . . . . 5  |-  ( x  =  X  ->  (
( x  .\/  y
)  e.  B  <->  ( X  .\/  y )  e.  B
) )
8 oveq1 6028 . . . . . 6  |-  ( x  =  X  ->  (
x  ./\  y )  =  ( X  ./\  y ) )
98eleq1d 2454 . . . . 5  |-  ( x  =  X  ->  (
( x  ./\  y
)  e.  B  <->  ( X  ./\  y )  e.  B
) )
107, 9anbi12d 692 . . . 4  |-  ( x  =  X  ->  (
( ( x  .\/  y )  e.  B  /\  ( x  ./\  y
)  e.  B )  <-> 
( ( X  .\/  y )  e.  B  /\  ( X  ./\  y
)  e.  B ) ) )
11 oveq2 6029 . . . . . 6  |-  ( y  =  Y  ->  ( X  .\/  y )  =  ( X  .\/  Y
) )
1211eleq1d 2454 . . . . 5  |-  ( y  =  Y  ->  (
( X  .\/  y
)  e.  B  <->  ( X  .\/  Y )  e.  B
) )
13 oveq2 6029 . . . . . 6  |-  ( y  =  Y  ->  ( X  ./\  y )  =  ( X  ./\  Y
) )
1413eleq1d 2454 . . . . 5  |-  ( y  =  Y  ->  (
( X  ./\  y
)  e.  B  <->  ( X  ./\ 
Y )  e.  B
) )
1512, 14anbi12d 692 . . . 4  |-  ( y  =  Y  ->  (
( ( X  .\/  y )  e.  B  /\  ( X  ./\  y
)  e.  B )  <-> 
( ( X  .\/  Y )  e.  B  /\  ( X  ./\  Y )  e.  B ) ) )
1610, 15rspc2v 3002 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. x  e.  B  A. y  e.  B  ( ( x 
.\/  y )  e.  B  /\  ( x 
./\  y )  e.  B )  ->  (
( X  .\/  Y
)  e.  B  /\  ( X  ./\  Y )  e.  B ) ) )
175, 16syl5com 28 . 2  |-  ( K  e.  Lat  ->  (
( X  e.  B  /\  Y  e.  B
)  ->  ( ( X  .\/  Y )  e.  B  /\  ( X 
./\  Y )  e.  B ) ) )
18173impib 1151 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .\/  Y )  e.  B  /\  ( X  ./\  Y )  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650   ` cfv 5395  (class class class)co 6021   Basecbs 13397   Posetcpo 14325   joincjn 14329   meetcmee 14330   Latclat 14402
This theorem is referenced by:  latjcl  14407  latmcl  14408
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
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 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-ov 6024  df-lat 14403
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