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Theorem dlatmjdi 14297
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 14296 . . 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 450 . 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 5865 . . . 4  |-  ( x  =  X  ->  (
x  ./\  ( y  .\/  z ) )  =  ( X  ./\  (
y  .\/  z )
) )
7 oveq1 5865 . . . . 5  |-  ( x  =  X  ->  (
x  ./\  y )  =  ( X  ./\  y ) )
8 oveq1 5865 . . . . 5  |-  ( x  =  X  ->  (
x  ./\  z )  =  ( X  ./\  z ) )
97, 8oveq12d 5876 . . . 4  |-  ( x  =  X  ->  (
( x  ./\  y
)  .\/  ( x  ./\  z ) )  =  ( ( X  ./\  y )  .\/  ( X  ./\  z ) ) )
106, 9eqeq12d 2297 . . 3  |-  ( x  =  X  ->  (
( x  ./\  (
y  .\/  z )
)  =  ( ( x  ./\  y )  .\/  ( x  ./\  z
) )  <->  ( X  ./\  ( y  .\/  z
) )  =  ( ( X  ./\  y
)  .\/  ( X  ./\  z ) ) ) )
11 oveq1 5865 . . . . 5  |-  ( y  =  Y  ->  (
y  .\/  z )  =  ( Y  .\/  z ) )
1211oveq2d 5874 . . . 4  |-  ( y  =  Y  ->  ( X  ./\  ( y  .\/  z ) )  =  ( X  ./\  ( Y  .\/  z ) ) )
13 oveq2 5866 . . . . 5  |-  ( y  =  Y  ->  ( X  ./\  y )  =  ( X  ./\  Y
) )
1413oveq1d 5873 . . . 4  |-  ( y  =  Y  ->  (
( X  ./\  y
)  .\/  ( X  ./\  z ) )  =  ( ( X  ./\  Y )  .\/  ( X 
./\  z ) ) )
1512, 14eqeq12d 2297 . . 3  |-  ( y  =  Y  ->  (
( X  ./\  (
y  .\/  z )
)  =  ( ( X  ./\  y )  .\/  ( X  ./\  z
) )  <->  ( X  ./\  ( Y  .\/  z
) )  =  ( ( X  ./\  Y
)  .\/  ( X  ./\  z ) ) ) )
16 oveq2 5866 . . . . 5  |-  ( z  =  Z  ->  ( Y  .\/  z )  =  ( Y  .\/  Z
) )
1716oveq2d 5874 . . . 4  |-  ( z  =  Z  ->  ( X  ./\  ( Y  .\/  z ) )  =  ( X  ./\  ( Y  .\/  Z ) ) )
18 oveq2 5866 . . . . 5  |-  ( z  =  Z  ->  ( X  ./\  z )  =  ( X  ./\  Z
) )
1918oveq2d 5874 . . . 4  |-  ( z  =  Z  ->  (
( X  ./\  Y
)  .\/  ( X  ./\  z ) )  =  ( ( X  ./\  Y )  .\/  ( X 
./\  Z ) ) )
2017, 19eqeq12d 2297 . . 3  |-  ( z  =  Z  ->  (
( X  ./\  ( Y  .\/  z ) )  =  ( ( X 
./\  Y )  .\/  ( X  ./\  z ) )  <->  ( X  ./\  ( Y  .\/  Z ) )  =  ( ( X  ./\  Y )  .\/  ( X  ./\  Z
) ) ) )
2110, 15, 20rspc3v 2893 . 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 455 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   joincjn 14078   meetcmee 14079   Latclat 14151  DLatcdlat 14294
This theorem is referenced by:  dlatjmdi  14300
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  ax-nul 4149
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-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-dlat 14295
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