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Theorem latlem 14469
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 14468 . . . 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 6080 . . . . . 6  |-  ( x  =  X  ->  (
x  .\/  y )  =  ( X  .\/  y ) )
76eleq1d 2501 . . . . 5  |-  ( x  =  X  ->  (
( x  .\/  y
)  e.  B  <->  ( X  .\/  y )  e.  B
) )
8 oveq1 6080 . . . . . 6  |-  ( x  =  X  ->  (
x  ./\  y )  =  ( X  ./\  y ) )
98eleq1d 2501 . . . . 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 6081 . . . . . 6  |-  ( y  =  Y  ->  ( X  .\/  y )  =  ( X  .\/  Y
) )
1211eleq1d 2501 . . . . 5  |-  ( y  =  Y  ->  (
( X  .\/  y
)  e.  B  <->  ( X  .\/  Y )  e.  B
) )
13 oveq2 6081 . . . . . 6  |-  ( y  =  Y  ->  ( X  ./\  y )  =  ( X  ./\  Y
) )
1413eleq1d 2501 . . . . 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 3050 . . 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 1652    e. wcel 1725   A.wral 2697   ` cfv 5446  (class class class)co 6073   Basecbs 13461   Posetcpo 14389   joincjn 14393   meetcmee 14394   Latclat 14466
This theorem is referenced by:  latjcl  14471  latmcl  14472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-lat 14467
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