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Theorem odudlatb 14551
Description: The dual of a distributive lattice is a distributive lattice and conversely. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Hypothesis
Ref Expression
odudlat.d  |-  D  =  (ODual `  K )
Assertion
Ref Expression
odudlatb  |-  ( K  e.  V  ->  ( K  e. DLat  <->  D  e. DLat ) )

Proof of Theorem odudlatb
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2389 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2389 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
3 eqid 2389 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
41, 2, 3latdisd 14545 . . . . 5  |-  ( K  e.  Lat  ->  ( A. x  e.  ( Base `  K ) A. y  e.  ( Base `  K ) A. z  e.  ( Base `  K
) ( x (
join `  K )
( y ( meet `  K ) z ) )  =  ( ( x ( join `  K
) y ) (
meet `  K )
( x ( join `  K ) z ) )  <->  A. x  e.  (
Base `  K ) A. y  e.  ( Base `  K ) A. z  e.  ( Base `  K ) ( x ( meet `  K
) ( y (
join `  K )
z ) )  =  ( ( x (
meet `  K )
y ) ( join `  K ) ( x ( meet `  K
) z ) ) ) )
54bicomd 193 . . . 4  |-  ( K  e.  Lat  ->  ( A. x  e.  ( Base `  K ) A. y  e.  ( Base `  K ) A. z  e.  ( Base `  K
) ( x (
meet `  K )
( y ( join `  K ) z ) )  =  ( ( x ( meet `  K
) y ) (
join `  K )
( x ( meet `  K ) z ) )  <->  A. x  e.  (
Base `  K ) A. y  e.  ( Base `  K ) A. z  e.  ( Base `  K ) ( x ( join `  K
) ( y (
meet `  K )
z ) )  =  ( ( x (
join `  K )
y ) ( meet `  K ) ( x ( join `  K
) z ) ) ) )
65pm5.32i 619 . . 3  |-  ( ( K  e.  Lat  /\  A. x  e.  ( Base `  K ) A. y  e.  ( Base `  K
) A. z  e.  ( Base `  K
) ( x (
meet `  K )
( y ( join `  K ) z ) )  =  ( ( x ( meet `  K
) y ) (
join `  K )
( x ( meet `  K ) z ) ) )  <->  ( K  e.  Lat  /\  A. x  e.  ( Base `  K
) A. y  e.  ( Base `  K
) A. z  e.  ( Base `  K
) ( x (
join `  K )
( y ( meet `  K ) z ) )  =  ( ( x ( join `  K
) y ) (
meet `  K )
( x ( join `  K ) z ) ) ) )
7 odudlat.d . . . . 5  |-  D  =  (ODual `  K )
87odulatb 14499 . . . 4  |-  ( K  e.  V  ->  ( K  e.  Lat  <->  D  e.  Lat ) )
98anbi1d 686 . . 3  |-  ( K  e.  V  ->  (
( K  e.  Lat  /\ 
A. x  e.  (
Base `  K ) A. y  e.  ( Base `  K ) A. z  e.  ( Base `  K ) ( x ( join `  K
) ( y (
meet `  K )
z ) )  =  ( ( x (
join `  K )
y ) ( meet `  K ) ( x ( join `  K
) z ) ) )  <->  ( D  e. 
Lat  /\  A. x  e.  ( Base `  K
) A. y  e.  ( Base `  K
) A. z  e.  ( Base `  K
) ( x (
join `  K )
( y ( meet `  K ) z ) )  =  ( ( x ( join `  K
) y ) (
meet `  K )
( x ( join `  K ) z ) ) ) ) )
106, 9syl5bb 249 . 2  |-  ( K  e.  V  ->  (
( K  e.  Lat  /\ 
A. x  e.  (
Base `  K ) A. y  e.  ( Base `  K ) A. z  e.  ( Base `  K ) ( x ( meet `  K
) ( y (
join `  K )
z ) )  =  ( ( x (
meet `  K )
y ) ( join `  K ) ( x ( meet `  K
) z ) ) )  <->  ( D  e. 
Lat  /\  A. x  e.  ( Base `  K
) A. y  e.  ( Base `  K
) A. z  e.  ( Base `  K
) ( x (
join `  K )
( y ( meet `  K ) z ) )  =  ( ( x ( join `  K
) y ) (
meet `  K )
( x ( join `  K ) z ) ) ) ) )
111, 2, 3isdlat 14548 . 2  |-  ( K  e. DLat 
<->  ( K  e.  Lat  /\ 
A. x  e.  (
Base `  K ) A. y  e.  ( Base `  K ) A. z  e.  ( Base `  K ) ( x ( meet `  K
) ( y (
join `  K )
z ) )  =  ( ( x (
meet `  K )
y ) ( join `  K ) ( x ( meet `  K
) z ) ) ) )
127, 1odubas 14489 . . 3  |-  ( Base `  K )  =  (
Base `  D )
137, 3odujoin 14498 . . 3  |-  ( meet `  K )  =  (
join `  D )
147, 2odumeet 14496 . . 3  |-  ( join `  K )  =  (
meet `  D )
1512, 13, 14isdlat 14548 . 2  |-  ( D  e. DLat 
<->  ( D  e.  Lat  /\ 
A. x  e.  (
Base `  K ) A. y  e.  ( Base `  K ) A. z  e.  ( Base `  K ) ( x ( join `  K
) ( y (
meet `  K )
z ) )  =  ( ( x (
join `  K )
y ) ( meet `  K ) ( x ( join `  K
) z ) ) ) )
1610, 11, 153bitr4g 280 1  |-  ( K  e.  V  ->  ( K  e. DLat  <->  D  e. DLat ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   ` cfv 5396  (class class class)co 6022   Basecbs 13398   joincjn 14330   meetcmee 14331   Latclat 14403  ODualcodu 14484  DLatcdlat 14546
This theorem is referenced by:  dlatjmdi  14552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ple 13478  df-poset 14332  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-lat 14404  df-odu 14485  df-dlat 14547
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