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Theorem latdisd 14309
Description: In a lattice, joins distribute over meets if and only if meets distribute over joins; the distributive property is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Hypotheses
Ref Expression
latdisd.b  |-  B  =  ( Base `  K
)
latdisd.j  |-  .\/  =  ( join `  K )
latdisd.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latdisd  |-  ( K  e.  Lat  ->  ( A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .\/  ( y 
./\  z ) )  =  ( ( x 
.\/  y )  ./\  ( x  .\/  z ) )  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  ./\  ( y 
.\/  z ) )  =  ( ( x 
./\  y )  .\/  ( x  ./\  z ) ) ) )
Distinct variable groups:    x, y,
z, K    x, B, y, z    x,  .\/ , y,
z    x,  ./\ , y, z

Proof of Theorem latdisd
Dummy variables  v  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 latdisd.b . . . 4  |-  B  =  ( Base `  K
)
2 latdisd.j . . . 4  |-  .\/  =  ( join `  K )
3 latdisd.m . . . 4  |-  ./\  =  ( meet `  K )
41, 2, 3latdisdlem 14308 . . 3  |-  ( K  e.  Lat  ->  ( A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .\/  ( y 
./\  z ) )  =  ( ( x 
.\/  y )  ./\  ( x  .\/  z ) )  ->  A. u  e.  B  A. v  e.  B  A. w  e.  B  ( u  ./\  ( v  .\/  w
) )  =  ( ( u  ./\  v
)  .\/  ( u  ./\  w ) ) ) )
5 eqid 2296 . . . . 5  |-  (ODual `  K )  =  (ODual `  K )
65odulat 14265 . . . 4  |-  ( K  e.  Lat  ->  (ODual `  K )  e.  Lat )
75, 1odubas 14253 . . . . 5  |-  B  =  ( Base `  (ODual `  K ) )
85, 3odujoin 14262 . . . . 5  |-  ./\  =  ( join `  (ODual `  K
) )
95, 2odumeet 14260 . . . . 5  |-  .\/  =  ( meet `  (ODual `  K
) )
107, 8, 9latdisdlem 14308 . . . 4  |-  ( (ODual `  K )  e.  Lat  ->  ( A. u  e.  B  A. v  e.  B  A. w  e.  B  ( u  ./\  ( v  .\/  w
) )  =  ( ( u  ./\  v
)  .\/  ( u  ./\  w ) )  ->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .\/  ( y 
./\  z ) )  =  ( ( x 
.\/  y )  ./\  ( x  .\/  z ) ) ) )
116, 10syl 15 . . 3  |-  ( K  e.  Lat  ->  ( A. u  e.  B  A. v  e.  B  A. w  e.  B  ( u  ./\  ( v 
.\/  w ) )  =  ( ( u 
./\  v )  .\/  ( u  ./\  w ) )  ->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .\/  ( y  ./\  z
) )  =  ( ( x  .\/  y
)  ./\  ( x  .\/  z ) ) ) )
124, 11impbid 183 . 2  |-  ( K  e.  Lat  ->  ( A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .\/  ( y 
./\  z ) )  =  ( ( x 
.\/  y )  ./\  ( x  .\/  z ) )  <->  A. u  e.  B  A. v  e.  B  A. w  e.  B  ( u  ./\  ( v 
.\/  w ) )  =  ( ( u 
./\  v )  .\/  ( u  ./\  w ) ) ) )
13 oveq1 5881 . . . 4  |-  ( u  =  x  ->  (
u  ./\  ( v  .\/  w ) )  =  ( x  ./\  (
v  .\/  w )
) )
14 oveq1 5881 . . . . 5  |-  ( u  =  x  ->  (
u  ./\  v )  =  ( x  ./\  v ) )
15 oveq1 5881 . . . . 5  |-  ( u  =  x  ->  (
u  ./\  w )  =  ( x  ./\  w ) )
1614, 15oveq12d 5892 . . . 4  |-  ( u  =  x  ->  (
( u  ./\  v
)  .\/  ( u  ./\  w ) )  =  ( ( x  ./\  v )  .\/  (
x  ./\  w )
) )
1713, 16eqeq12d 2310 . . 3  |-  ( u  =  x  ->  (
( u  ./\  (
v  .\/  w )
)  =  ( ( u  ./\  v )  .\/  ( u  ./\  w
) )  <->  ( x  ./\  ( v  .\/  w
) )  =  ( ( x  ./\  v
)  .\/  ( x  ./\  w ) ) ) )
18 oveq1 5881 . . . . 5  |-  ( v  =  y  ->  (
v  .\/  w )  =  ( y  .\/  w ) )
1918oveq2d 5890 . . . 4  |-  ( v  =  y  ->  (
x  ./\  ( v  .\/  w ) )  =  ( x  ./\  (
y  .\/  w )
) )
20 oveq2 5882 . . . . 5  |-  ( v  =  y  ->  (
x  ./\  v )  =  ( x  ./\  y ) )
2120oveq1d 5889 . . . 4  |-  ( v  =  y  ->  (
( x  ./\  v
)  .\/  ( x  ./\  w ) )  =  ( ( x  ./\  y )  .\/  (
x  ./\  w )
) )
2219, 21eqeq12d 2310 . . 3  |-  ( v  =  y  ->  (
( x  ./\  (
v  .\/  w )
)  =  ( ( x  ./\  v )  .\/  ( x  ./\  w
) )  <->  ( x  ./\  ( y  .\/  w
) )  =  ( ( x  ./\  y
)  .\/  ( x  ./\  w ) ) ) )
23 oveq2 5882 . . . . 5  |-  ( w  =  z  ->  (
y  .\/  w )  =  ( y  .\/  z ) )
2423oveq2d 5890 . . . 4  |-  ( w  =  z  ->  (
x  ./\  ( y  .\/  w ) )  =  ( x  ./\  (
y  .\/  z )
) )
25 oveq2 5882 . . . . 5  |-  ( w  =  z  ->  (
x  ./\  w )  =  ( x  ./\  z ) )
2625oveq2d 5890 . . . 4  |-  ( w  =  z  ->  (
( x  ./\  y
)  .\/  ( x  ./\  w ) )  =  ( ( x  ./\  y )  .\/  (
x  ./\  z )
) )
2724, 26eqeq12d 2310 . . 3  |-  ( w  =  z  ->  (
( x  ./\  (
y  .\/  w )
)  =  ( ( x  ./\  y )  .\/  ( x  ./\  w
) )  <->  ( x  ./\  ( y  .\/  z
) )  =  ( ( x  ./\  y
)  .\/  ( x  ./\  z ) ) ) )
2817, 22, 27cbvral3v 2787 . 2  |-  ( A. u  e.  B  A. v  e.  B  A. w  e.  B  (
u  ./\  ( v  .\/  w ) )  =  ( ( u  ./\  v )  .\/  (
u  ./\  w )
)  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  ./\  ( y 
.\/  z ) )  =  ( ( x 
./\  y )  .\/  ( x  ./\  z ) ) )
2912, 28syl6bb 252 1  |-  ( K  e.  Lat  ->  ( A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .\/  ( y 
./\  z ) )  =  ( ( x 
.\/  y )  ./\  ( x  .\/  z ) )  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  ./\  ( y 
.\/  z ) )  =  ( ( x 
./\  y )  .\/  ( x  ./\  z ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   joincjn 14094   meetcmee 14095   Latclat 14167  ODualcodu 14248
This theorem is referenced by:  odudlatb  14315
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ple 13244  df-poset 14096  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-lat 14168  df-odu 14249
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