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Theorem latdisd 14608
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 14607 . . 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 2435 . . . . 5  |-  (ODual `  K )  =  (ODual `  K )
65odulat 14564 . . . 4  |-  ( K  e.  Lat  ->  (ODual `  K )  e.  Lat )
75, 1odubas 14552 . . . . 5  |-  B  =  ( Base `  (ODual `  K ) )
85, 3odujoin 14561 . . . . 5  |-  ./\  =  ( join `  (ODual `  K
) )
95, 2odumeet 14559 . . . . 5  |-  .\/  =  ( meet `  (ODual `  K
) )
107, 8, 9latdisdlem 14607 . . . 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 16 . . 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 184 . 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 6080 . . . 4  |-  ( u  =  x  ->  (
u  ./\  ( v  .\/  w ) )  =  ( x  ./\  (
v  .\/  w )
) )
14 oveq1 6080 . . . . 5  |-  ( u  =  x  ->  (
u  ./\  v )  =  ( x  ./\  v ) )
15 oveq1 6080 . . . . 5  |-  ( u  =  x  ->  (
u  ./\  w )  =  ( x  ./\  w ) )
1614, 15oveq12d 6091 . . . 4  |-  ( u  =  x  ->  (
( u  ./\  v
)  .\/  ( u  ./\  w ) )  =  ( ( x  ./\  v )  .\/  (
x  ./\  w )
) )
1713, 16eqeq12d 2449 . . 3  |-  ( u  =  x  ->  (
( u  ./\  (
v  .\/  w )
)  =  ( ( u  ./\  v )  .\/  ( u  ./\  w
) )  <->  ( x  ./\  ( v  .\/  w
) )  =  ( ( x  ./\  v
)  .\/  ( x  ./\  w ) ) ) )
18 oveq1 6080 . . . . 5  |-  ( v  =  y  ->  (
v  .\/  w )  =  ( y  .\/  w ) )
1918oveq2d 6089 . . . 4  |-  ( v  =  y  ->  (
x  ./\  ( v  .\/  w ) )  =  ( x  ./\  (
y  .\/  w )
) )
20 oveq2 6081 . . . . 5  |-  ( v  =  y  ->  (
x  ./\  v )  =  ( x  ./\  y ) )
2120oveq1d 6088 . . . 4  |-  ( v  =  y  ->  (
( x  ./\  v
)  .\/  ( x  ./\  w ) )  =  ( ( x  ./\  y )  .\/  (
x  ./\  w )
) )
2219, 21eqeq12d 2449 . . 3  |-  ( v  =  y  ->  (
( x  ./\  (
v  .\/  w )
)  =  ( ( x  ./\  v )  .\/  ( x  ./\  w
) )  <->  ( x  ./\  ( y  .\/  w
) )  =  ( ( x  ./\  y
)  .\/  ( x  ./\  w ) ) ) )
23 oveq2 6081 . . . . 5  |-  ( w  =  z  ->  (
y  .\/  w )  =  ( y  .\/  z ) )
2423oveq2d 6089 . . . 4  |-  ( w  =  z  ->  (
x  ./\  ( y  .\/  w ) )  =  ( x  ./\  (
y  .\/  z )
) )
25 oveq2 6081 . . . . 5  |-  ( w  =  z  ->  (
x  ./\  w )  =  ( x  ./\  z ) )
2625oveq2d 6089 . . . 4  |-  ( w  =  z  ->  (
( x  ./\  y
)  .\/  ( x  ./\  w ) )  =  ( ( x  ./\  y )  .\/  (
x  ./\  z )
) )
2724, 26eqeq12d 2449 . . 3  |-  ( w  =  z  ->  (
( x  ./\  (
y  .\/  w )
)  =  ( ( x  ./\  y )  .\/  ( x  ./\  w
) )  <->  ( x  ./\  ( y  .\/  z
) )  =  ( ( x  ./\  y
)  .\/  ( x  ./\  z ) ) ) )
2817, 22, 27cbvral3v 2934 . 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 253 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 177    = wceq 1652    e. wcel 1725   A.wral 2697   ` cfv 5446  (class class class)co 6073   Basecbs 13461   joincjn 14393   meetcmee 14394   Latclat 14466  ODualcodu 14547
This theorem is referenced by:  odudlatb  14614
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ple 13541  df-poset 14395  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-lat 14467  df-odu 14548
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