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Theorem dlatmjdi 14612
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 14611 . . 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 6080 . . . 4  |-  ( x  =  X  ->  (
x  ./\  ( y  .\/  z ) )  =  ( X  ./\  (
y  .\/  z )
) )
7 oveq1 6080 . . . . 5  |-  ( x  =  X  ->  (
x  ./\  y )  =  ( X  ./\  y ) )
8 oveq1 6080 . . . . 5  |-  ( x  =  X  ->  (
x  ./\  z )  =  ( X  ./\  z ) )
97, 8oveq12d 6091 . . . 4  |-  ( x  =  X  ->  (
( x  ./\  y
)  .\/  ( x  ./\  z ) )  =  ( ( X  ./\  y )  .\/  ( X  ./\  z ) ) )
106, 9eqeq12d 2449 . . 3  |-  ( x  =  X  ->  (
( x  ./\  (
y  .\/  z )
)  =  ( ( x  ./\  y )  .\/  ( x  ./\  z
) )  <->  ( X  ./\  ( y  .\/  z
) )  =  ( ( X  ./\  y
)  .\/  ( X  ./\  z ) ) ) )
11 oveq1 6080 . . . . 5  |-  ( y  =  Y  ->  (
y  .\/  z )  =  ( Y  .\/  z ) )
1211oveq2d 6089 . . . 4  |-  ( y  =  Y  ->  ( X  ./\  ( y  .\/  z ) )  =  ( X  ./\  ( Y  .\/  z ) ) )
13 oveq2 6081 . . . . 5  |-  ( y  =  Y  ->  ( X  ./\  y )  =  ( X  ./\  Y
) )
1413oveq1d 6088 . . . 4  |-  ( y  =  Y  ->  (
( X  ./\  y
)  .\/  ( X  ./\  z ) )  =  ( ( X  ./\  Y )  .\/  ( X 
./\  z ) ) )
1512, 14eqeq12d 2449 . . 3  |-  ( y  =  Y  ->  (
( X  ./\  (
y  .\/  z )
)  =  ( ( X  ./\  y )  .\/  ( X  ./\  z
) )  <->  ( X  ./\  ( Y  .\/  z
) )  =  ( ( X  ./\  Y
)  .\/  ( X  ./\  z ) ) ) )
16 oveq2 6081 . . . . 5  |-  ( z  =  Z  ->  ( Y  .\/  z )  =  ( Y  .\/  Z
) )
1716oveq2d 6089 . . . 4  |-  ( z  =  Z  ->  ( X  ./\  ( Y  .\/  z ) )  =  ( X  ./\  ( Y  .\/  Z ) ) )
18 oveq2 6081 . . . . 5  |-  ( z  =  Z  ->  ( X  ./\  z )  =  ( X  ./\  Z
) )
1918oveq2d 6089 . . . 4  |-  ( z  =  Z  ->  (
( X  ./\  Y
)  .\/  ( X  ./\  z ) )  =  ( ( X  ./\  Y )  .\/  ( X 
./\  Z ) ) )
2017, 19eqeq12d 2449 . . 3  |-  ( z  =  Z  ->  (
( X  ./\  ( Y  .\/  z ) )  =  ( ( X 
./\  Y )  .\/  ( X  ./\  z ) )  <->  ( X  ./\  ( Y  .\/  Z ) )  =  ( ( X  ./\  Y )  .\/  ( X  ./\  Z
) ) ) )
2110, 15, 20rspc3v 3053 . 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 1652    e. wcel 1725   A.wral 2697   ` cfv 5446  (class class class)co 6073   Basecbs 13461   joincjn 14393   meetcmee 14394   Latclat 14466  DLatcdlat 14609
This theorem is referenced by:  dlatjmdi  14615
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  ax-nul 4330
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-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  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-dlat 14610
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