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Theorem dlatmjdi 14548
Description: In a distributive lattice, meets distribute over joins. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Hypotheses
Ref Expression
isdlat.b  |-  B  =  ( Base `  K
)
isdlat.j  |-  .\/  =  ( join `  K )
isdlat.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
dlatmjdi  |-  ( ( K  e. DLat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  ./\  ( Y  .\/  Z
) )  =  ( ( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) ) )

Proof of Theorem dlatmjdi
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isdlat.b . . . 4  |-  B  =  ( Base `  K
)
2 isdlat.j . . . 4  |-  .\/  =  ( join `  K )
3 isdlat.m . . . 4  |-  ./\  =  ( meet `  K )
41, 2, 3isdlat 14547 . . 3  |-  ( K  e. DLat 
<->  ( K  e.  Lat  /\ 
A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  ./\  ( y 
.\/  z ) )  =  ( ( x 
./\  y )  .\/  ( x  ./\  z ) ) ) )
54simprbi 451 . 2  |-  ( K  e. DLat  ->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  ./\  ( y 
.\/  z ) )  =  ( ( x 
./\  y )  .\/  ( x  ./\  z ) ) )
6 oveq1 6028 . . . 4  |-  ( x  =  X  ->  (
x  ./\  ( y  .\/  z ) )  =  ( X  ./\  (
y  .\/  z )
) )
7 oveq1 6028 . . . . 5  |-  ( x  =  X  ->  (
x  ./\  y )  =  ( X  ./\  y ) )
8 oveq1 6028 . . . . 5  |-  ( x  =  X  ->  (
x  ./\  z )  =  ( X  ./\  z ) )
97, 8oveq12d 6039 . . . 4  |-  ( x  =  X  ->  (
( x  ./\  y
)  .\/  ( x  ./\  z ) )  =  ( ( X  ./\  y )  .\/  ( X  ./\  z ) ) )
106, 9eqeq12d 2402 . . 3  |-  ( x  =  X  ->  (
( x  ./\  (
y  .\/  z )
)  =  ( ( x  ./\  y )  .\/  ( x  ./\  z
) )  <->  ( X  ./\  ( y  .\/  z
) )  =  ( ( X  ./\  y
)  .\/  ( X  ./\  z ) ) ) )
11 oveq1 6028 . . . . 5  |-  ( y  =  Y  ->  (
y  .\/  z )  =  ( Y  .\/  z ) )
1211oveq2d 6037 . . . 4  |-  ( y  =  Y  ->  ( X  ./\  ( y  .\/  z ) )  =  ( X  ./\  ( Y  .\/  z ) ) )
13 oveq2 6029 . . . . 5  |-  ( y  =  Y  ->  ( X  ./\  y )  =  ( X  ./\  Y
) )
1413oveq1d 6036 . . . 4  |-  ( y  =  Y  ->  (
( X  ./\  y
)  .\/  ( X  ./\  z ) )  =  ( ( X  ./\  Y )  .\/  ( X 
./\  z ) ) )
1512, 14eqeq12d 2402 . . 3  |-  ( y  =  Y  ->  (
( X  ./\  (
y  .\/  z )
)  =  ( ( X  ./\  y )  .\/  ( X  ./\  z
) )  <->  ( X  ./\  ( Y  .\/  z
) )  =  ( ( X  ./\  Y
)  .\/  ( X  ./\  z ) ) ) )
16 oveq2 6029 . . . . 5  |-  ( z  =  Z  ->  ( Y  .\/  z )  =  ( Y  .\/  Z
) )
1716oveq2d 6037 . . . 4  |-  ( z  =  Z  ->  ( X  ./\  ( Y  .\/  z ) )  =  ( X  ./\  ( Y  .\/  Z ) ) )
18 oveq2 6029 . . . . 5  |-  ( z  =  Z  ->  ( X  ./\  z )  =  ( X  ./\  Z
) )
1918oveq2d 6037 . . . 4  |-  ( z  =  Z  ->  (
( X  ./\  Y
)  .\/  ( X  ./\  z ) )  =  ( ( X  ./\  Y )  .\/  ( X 
./\  Z ) ) )
2017, 19eqeq12d 2402 . . 3  |-  ( z  =  Z  ->  (
( X  ./\  ( Y  .\/  z ) )  =  ( ( X 
./\  Y )  .\/  ( X  ./\  z ) )  <->  ( X  ./\  ( Y  .\/  Z ) )  =  ( ( X  ./\  Y )  .\/  ( X  ./\  Z
) ) ) )
2110, 15, 20rspc3v 3005 . 2  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  ./\  ( y  .\/  z
) )  =  ( ( x  ./\  y
)  .\/  ( x  ./\  z ) )  -> 
( X  ./\  ( Y  .\/  Z ) )  =  ( ( X 
./\  Y )  .\/  ( X  ./\  Z ) ) ) )
225, 21mpan9 456 1  |-  ( ( K  e. DLat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  ./\  ( Y  .\/  Z
) )  =  ( ( X  ./\  Y
)  .\/  ( X  ./\ 
Z ) ) )
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   joincjn 14329   meetcmee 14330   Latclat 14402  DLatcdlat 14545
This theorem is referenced by:  dlatjmdi  14551
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  ax-nul 4280
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-eu 2243  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  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-dlat 14546
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