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.

Step	Hyp	Ref	Expression
1		odupos.d	. . . 4 |- D = (ODual ` O )
2		fvex 5804	. . . 4 |- (ODual ` O ) e. _V
3	1, 2	eqeltri 2536	. . 3 |- D e. _V
4	3	a1i 11	. 2 |- ( O e. Poset -> D e. _V )
5		eqid 2452	. . . 4 |- ( Base ` O ) = ( Base ` O )
6	1, 5	odubas 15417	. . 3 |- ( Base ` O ) = ( Base ` D )
7	6	a1i 11	. 2 |- ( O e. Poset -> ( Base ` O ) = ( Base ` D ) )
8		eqid 2452	. . . 4 |- ( le ` O ) = ( le ` O )
9	1, 8	oduleval 15415	. . 3 |- `' ( le ` O ) = ( le ` D )
10	9	a1i 11	. 2 |- ( O e. Poset -> `' ( le ` O ) = ( le ` D ) )
11	5, 8	posref 15235	. . 3 |- ( ( O e. Poset /\ a e. ( Base ` O ) ) -> a ( le ` O ) a )
12		vex 3075	. . . 4 |- a e. _V
13	12, 12	brcnv 5125	. . 3 |- ( a `' ( le ` O ) a <-> a ( le ` O ) a )
14	11, 13	sylibr 212	. 2 |- ( ( O e. Poset /\ a e. ( Base ` O ) ) -> a `' ( le ` O ) a )
15		vex 3075	. . . . 5 |- b e. _V
16	12, 15	brcnv 5125	. . . 4 |- ( a `' ( le ` O ) b <-> b ( le ` O ) a )
17	15, 12	brcnv 5125	. . . 4 |- ( b `' ( le ` O ) a <-> a ( le ` O ) b )
18	16, 17	anbi12ci 698	. . 3 |- ( ( a `' ( le ` O ) b /\ b `' ( le ` O ) a ) <-> ( a ( le ` O ) b /\ b ( le ` O ) a ) )
19	5, 8	posasymb 15236	. . . 4 |- ( ( O e. Poset /\ a e. ( Base ` O ) /\ b e. ( Base ` O ) ) -> ( ( a ( le ` O ) b /\ b ( le ` O ) a ) <-> a = b ) )
20	19	biimpd 207	. . 3 |- ( ( O e. Poset /\ a e. ( Base ` O ) /\ b e. ( Base ` O ) ) -> ( ( a ( le ` O ) b /\ b ( le ` O ) a ) -> a = b ) )
21	18, 20	syl5bi 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 ) ) )
23	5, 8	postr 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 ) )
24	22, 23	sylan2b 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
26	15, 25	brcnv 5125	. . . 4 |- ( b `' ( le ` O ) c <-> c ( le ` O ) b )
27	16, 26	anbi12ci 698	. . 3 |- ( ( a `' ( le ` O ) b /\ b `' ( le ` O ) c ) <-> ( c ( le ` O ) b /\ b ( le ` O ) a ) )
28	12, 25	brcnv 5125	. . 3 |- ( a `' ( le ` O ) c <-> c ( le ` O ) a )
29	24, 27, 28	3imtr4g 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 ) )
30	4, 7, 10, 14, 21, 29	isposd 15239	1 |- ( O e. Poset -> D e. Poset )

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