MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dlatmjdi Unicode version

Theorem dlatmjdi 14313
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 14312 . . 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 5881 . . . 4  |-  ( x  =  X  ->  (
x  ./\  ( y  .\/  z ) )  =  ( X  ./\  (
y  .\/  z )
) )
7 oveq1 5881 . . . . 5  |-  ( x  =  X  ->  (
x  ./\  y )  =  ( X  ./\  y ) )
8 oveq1 5881 . . . . 5  |-  ( x  =  X  ->  (
x  ./\  z )  =  ( X  ./\  z ) )
97, 8oveq12d 5892 . . . 4  |-  ( x  =  X  ->  (
( x  ./\  y
)  .\/  ( x  ./\  z ) )  =  ( ( X  ./\  y )  .\/  ( X  ./\  z ) ) )
106, 9eqeq12d 2310 . . 3  |-  ( x  =  X  ->  (
( x  ./\  (
y  .\/  z )
)  =  ( ( x  ./\  y )  .\/  ( x  ./\  z
) )  <->  ( X  ./\  ( y  .\/  z
) )  =  ( ( X  ./\  y
)  .\/  ( X  ./\  z ) ) ) )
11 oveq1 5881 . . . . 5  |-  ( y  =  Y  ->  (
y  .\/  z )  =  ( Y  .\/  z ) )
1211oveq2d 5890 . . . 4  |-  ( y  =  Y  ->  ( X  ./\  ( y  .\/  z ) )  =  ( X  ./\  ( Y  .\/  z ) ) )
13 oveq2 5882 . . . . 5  |-  ( y  =  Y  ->  ( X  ./\  y )  =  ( X  ./\  Y
) )
1413oveq1d 5889 . . . 4  |-  ( y  =  Y  ->  (
( X  ./\  y
)  .\/  ( X  ./\  z ) )  =  ( ( X  ./\  Y )  .\/  ( X 
./\  z ) ) )
1512, 14eqeq12d 2310 . . 3  |-  ( y  =  Y  ->  (
( X  ./\  (
y  .\/  z )
)  =  ( ( X  ./\  y )  .\/  ( X  ./\  z
) )  <->  ( X  ./\  ( Y  .\/  z
) )  =  ( ( X  ./\  Y
)  .\/  ( X  ./\  z ) ) ) )
16 oveq2 5882 . . . . 5  |-  ( z  =  Z  ->  ( Y  .\/  z )  =  ( Y  .\/  Z
) )
1716oveq2d 5890 . . . 4  |-  ( z  =  Z  ->  ( X  ./\  ( Y  .\/  z ) )  =  ( X  ./\  ( Y  .\/  Z ) ) )
18 oveq2 5882 . . . . 5  |-  ( z  =  Z  ->  ( X  ./\  z )  =  ( X  ./\  Z
) )
1918oveq2d 5890 . . . 4  |-  ( z  =  Z  ->  (
( X  ./\  Y
)  .\/  ( X  ./\  z ) )  =  ( ( X  ./\  Y )  .\/  ( X 
./\  Z ) ) )
2017, 19eqeq12d 2310 . . 3  |-  ( z  =  Z  ->  (
( X  ./\  ( Y  .\/  z ) )  =  ( ( X 
./\  Y )  .\/  ( X  ./\  z ) )  <->  ( X  ./\  ( Y  .\/  Z ) )  =  ( ( X  ./\  Y )  .\/  ( X  ./\  Z
) ) ) )
2110, 15, 20rspc3v 2906 . 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 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   joincjn 14094   meetcmee 14095   Latclat 14167  DLatcdlat 14310
This theorem is referenced by:  dlatjmdi  14316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-dlat 14311
  Copyright terms: Public domain W3C validator