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Theorem odupos 15419
Description: Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Hypothesis
Ref Expression
odupos.d  |-  D  =  (ODual `  O )
Assertion
Ref Expression
odupos  |-  ( O  e.  Poset  ->  D  e.  Poset
)

Proof of Theorem odupos
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 odupos.d . . . 4  |-  D  =  (ODual `  O )
2 fvex 5804 . . . 4  |-  (ODual `  O )  e.  _V
31, 2eqeltri 2536 . . 3  |-  D  e. 
_V
43a1i 11 . 2  |-  ( O  e.  Poset  ->  D  e.  _V )
5 eqid 2452 . . . 4  |-  ( Base `  O )  =  (
Base `  O )
61, 5odubas 15417 . . 3  |-  ( Base `  O )  =  (
Base `  D )
76a1i 11 . 2  |-  ( O  e.  Poset  ->  ( Base `  O )  =  (
Base `  D )
)
8 eqid 2452 . . . 4  |-  ( le
`  O )  =  ( le `  O
)
91, 8oduleval 15415 . . 3  |-  `' ( le `  O )  =  ( le `  D )
109a1i 11 . 2  |-  ( O  e.  Poset  ->  `' ( le `  O )  =  ( le `  D
) )
115, 8posref 15235 . . 3  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
) )  ->  a
( le `  O
) a )
12 vex 3075 . . . 4  |-  a  e. 
_V
1312, 12brcnv 5125 . . 3  |-  ( a `' ( le `  O ) a  <->  a ( le `  O ) a )
1411, 13sylibr 212 . 2  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
) )  ->  a `' ( le `  O ) a )
15 vex 3075 . . . . 5  |-  b  e. 
_V
1612, 15brcnv 5125 . . . 4  |-  ( a `' ( le `  O ) b  <->  b ( le `  O ) a )
1715, 12brcnv 5125 . . . 4  |-  ( b `' ( le `  O ) a  <->  a ( le `  O ) b )
1816, 17anbi12ci 698 . . 3  |-  ( ( a `' ( le
`  O ) b  /\  b `' ( le `  O ) a )  <->  ( a
( le `  O
) b  /\  b
( le `  O
) a ) )
195, 8posasymb 15236 . . . 4  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )
)  ->  ( (
a ( le `  O ) b  /\  b ( le `  O ) a )  <-> 
a  =  b ) )
2019biimpd 207 . . 3  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )
)  ->  ( (
a ( le `  O ) b  /\  b ( le `  O ) a )  ->  a  =  b ) )
2118, 20syl5bi 217 . 2  |-  ( ( O  e.  Poset  /\  a  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )
)  ->  ( (
a `' ( le
`  O ) b  /\  b `' ( le `  O ) a )  ->  a  =  b ) )
22 3anrev 976 . . . 4  |-  ( ( a  e.  ( Base `  O )  /\  b  e.  ( Base `  O
)  /\  c  e.  ( Base `  O )
)  <->  ( c  e.  ( Base `  O
)  /\  b  e.  ( Base `  O )  /\  a  e.  ( Base `  O ) ) )
235, 8postr 15237 . . . 4  |-  ( ( O  e.  Poset  /\  (
c  e.  ( Base `  O )  /\  b  e.  ( Base `  O
)  /\  a  e.  ( Base `  O )
) )  ->  (
( c ( le
`  O ) b  /\  b ( le
`  O ) a )  ->  c ( le `  O ) a ) )
2422, 23sylan2b 475 . . 3  |-  ( ( O  e.  Poset  /\  (
a  e.  ( Base `  O )  /\  b  e.  ( Base `  O
)  /\  c  e.  ( Base `  O )
) )  ->  (
( c ( le
`  O ) b  /\  b ( le
`  O ) a )  ->  c ( le `  O ) a ) )
25 vex 3075 . . . . 5  |-  c  e. 
_V
2615, 25brcnv 5125 . . . 4  |-  ( b `' ( le `  O ) c  <->  c ( le `  O ) b )
2716, 26anbi12ci 698 . . 3  |-  ( ( a `' ( le
`  O ) b  /\  b `' ( le `  O ) c )  <->  ( c
( le `  O
) b  /\  b
( le `  O
) a ) )
2812, 25brcnv 5125 . . 3  |-  ( a `' ( le `  O ) c  <->  c ( le `  O ) a )
2924, 27, 283imtr4g 270 . 2  |-  ( ( O  e.  Poset  /\  (
a  e.  ( Base `  O )  /\  b  e.  ( Base `  O
)  /\  c  e.  ( Base `  O )
) )  ->  (
( a `' ( le `  O ) b  /\  b `' ( le `  O
) c )  -> 
a `' ( le
`  O ) c ) )
304, 7, 10, 14, 21, 29isposd 15239 1  |-  ( O  e.  Poset  ->  D  e.  Poset
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3072   class class class wbr 4395   `'ccnv 4942   ` cfv 5521   Basecbs 14287   lecple 14359   Posetcpo 15224  ODualcodu 15412
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 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ple 14372  df-poset 15230  df-odu 15413
This theorem is referenced by:  oduposb  15420  posglbd  15434  odutos  26264
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