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Theorem latlem 14154
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 14153 . . . 4  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  A. x  e.  B  A. y  e.  B  ( (
x  .\/  y )  e.  B  /\  (
x  ./\  y )  e.  B ) ) )
54simprbi 450 . . 3  |-  ( K  e.  Lat  ->  A. x  e.  B  A. y  e.  B  ( (
x  .\/  y )  e.  B  /\  (
x  ./\  y )  e.  B ) )
6 oveq1 5865 . . . . . 6  |-  ( x  =  X  ->  (
x  .\/  y )  =  ( X  .\/  y ) )
76eleq1d 2349 . . . . 5  |-  ( x  =  X  ->  (
( x  .\/  y
)  e.  B  <->  ( X  .\/  y )  e.  B
) )
8 oveq1 5865 . . . . . 6  |-  ( x  =  X  ->  (
x  ./\  y )  =  ( X  ./\  y ) )
98eleq1d 2349 . . . . 5  |-  ( x  =  X  ->  (
( x  ./\  y
)  e.  B  <->  ( X  ./\  y )  e.  B
) )
107, 9anbi12d 691 . . . 4  |-  ( x  =  X  ->  (
( ( x  .\/  y )  e.  B  /\  ( x  ./\  y
)  e.  B )  <-> 
( ( X  .\/  y )  e.  B  /\  ( X  ./\  y
)  e.  B ) ) )
11 oveq2 5866 . . . . . 6  |-  ( y  =  Y  ->  ( X  .\/  y )  =  ( X  .\/  Y
) )
1211eleq1d 2349 . . . . 5  |-  ( y  =  Y  ->  (
( X  .\/  y
)  e.  B  <->  ( X  .\/  Y )  e.  B
) )
13 oveq2 5866 . . . . . 6  |-  ( y  =  Y  ->  ( X  ./\  y )  =  ( X  ./\  Y
) )
1413eleq1d 2349 . . . . 5  |-  ( y  =  Y  ->  (
( X  ./\  y
)  e.  B  <->  ( X  ./\ 
Y )  e.  B
) )
1512, 14anbi12d 691 . . . 4  |-  ( y  =  Y  ->  (
( ( X  .\/  y )  e.  B  /\  ( X  ./\  y
)  e.  B )  <-> 
( ( X  .\/  Y )  e.  B  /\  ( X  ./\  Y )  e.  B ) ) )
1610, 15rspc2v 2890 . . 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 26 . 2  |-  ( K  e.  Lat  ->  (
( X  e.  B  /\  Y  e.  B
)  ->  ( ( X  .\/  Y )  e.  B  /\  ( X 
./\  Y )  e.  B ) ) )
18173impib 1149 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 358    /\ w3a 934    = 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 is referenced by:  latjcl  14156  latmcl  14157
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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