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Theorem odulatb 15417
Description: Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Hypothesis
Ref Expression
oduglb.d  |-  D  =  (ODual `  O )
Assertion
Ref Expression
odulatb  |-  ( O  e.  V  ->  ( O  e.  Lat  <->  D  e.  Lat ) )

Proof of Theorem odulatb
StepHypRef Expression
1 oduglb.d . . . 4  |-  D  =  (ODual `  O )
21oduposb 15410 . . 3  |-  ( O  e.  V  ->  ( O  e.  Poset  <->  D  e.  Poset
) )
3 ancom 450 . . . 4  |-  ( ( dom  ( join `  O
)  =  ( (
Base `  O )  X.  ( Base `  O
) )  /\  dom  ( meet `  O )  =  ( ( Base `  O )  X.  ( Base `  O ) ) )  <->  ( dom  ( meet `  O )  =  ( ( Base `  O
)  X.  ( Base `  O ) )  /\  dom  ( join `  O
)  =  ( (
Base `  O )  X.  ( Base `  O
) ) ) )
43a1i 11 . . 3  |-  ( O  e.  V  ->  (
( dom  ( join `  O )  =  ( ( Base `  O
)  X.  ( Base `  O ) )  /\  dom  ( meet `  O
)  =  ( (
Base `  O )  X.  ( Base `  O
) ) )  <->  ( dom  ( meet `  O )  =  ( ( Base `  O )  X.  ( Base `  O ) )  /\  dom  ( join `  O )  =  ( ( Base `  O
)  X.  ( Base `  O ) ) ) ) )
52, 4anbi12d 710 . 2  |-  ( O  e.  V  ->  (
( O  e.  Poset  /\  ( dom  ( join `  O )  =  ( ( Base `  O
)  X.  ( Base `  O ) )  /\  dom  ( meet `  O
)  =  ( (
Base `  O )  X.  ( Base `  O
) ) ) )  <-> 
( D  e.  Poset  /\  ( dom  ( meet `  O )  =  ( ( Base `  O
)  X.  ( Base `  O ) )  /\  dom  ( join `  O
)  =  ( (
Base `  O )  X.  ( Base `  O
) ) ) ) ) )
6 eqid 2451 . . 3  |-  ( Base `  O )  =  (
Base `  O )
7 eqid 2451 . . 3  |-  ( join `  O )  =  (
join `  O )
8 eqid 2451 . . 3  |-  ( meet `  O )  =  (
meet `  O )
96, 7, 8islat 15321 . 2  |-  ( O  e.  Lat  <->  ( O  e.  Poset  /\  ( dom  ( join `  O )  =  ( ( Base `  O )  X.  ( Base `  O ) )  /\  dom  ( meet `  O )  =  ( ( Base `  O
)  X.  ( Base `  O ) ) ) ) )
101, 6odubas 15407 . . 3  |-  ( Base `  O )  =  (
Base `  D )
111, 8odujoin 15416 . . 3  |-  ( meet `  O )  =  (
join `  D )
121, 7odumeet 15414 . . 3  |-  ( join `  O )  =  (
meet `  D )
1310, 11, 12islat 15321 . 2  |-  ( D  e.  Lat  <->  ( D  e.  Poset  /\  ( dom  ( meet `  O )  =  ( ( Base `  O )  X.  ( Base `  O ) )  /\  dom  ( join `  O )  =  ( ( Base `  O
)  X.  ( Base `  O ) ) ) ) )
145, 9, 133bitr4g 288 1  |-  ( O  e.  V  ->  ( O  e.  Lat  <->  D  e.  Lat ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    X. cxp 4938   dom cdm 4940   ` cfv 5518   Basecbs 14278   Posetcpo 15214   joincjn 15218   meetcmee 15219   Latclat 15319  ODualcodu 15402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ple 14362  df-poset 15220  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-lat 15320  df-odu 15403
This theorem is referenced by:  odulat  15419  odudlatb  15470
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