Theorem odupos 16459

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 5889	. . . 4 |- (ODual ` O ) e. _V
3	1, 2	eqeltri 2545	. . 3 |- D e. _V
4	3	a1i 11	. 2 |- ( O e. Poset -> D e. _V )
5		eqid 2471	. . . 4 |- ( Base ` O ) = ( Base ` O )
6	1, 5	odubas 16457	. . 3 |- ( Base ` O ) = ( Base ` D )
7	6	a1i 11	. 2 |- ( O e. Poset -> ( Base ` O ) = ( Base ` D ) )
8		eqid 2471	. . . 4 |- ( le ` O ) = ( le ` O )
9	1, 8	oduleval 16455	. . 3 |- `' ( le ` O ) = ( le ` D )
10	9	a1i 11	. 2 |- ( O e. Poset -> `' ( le ` O ) = ( le ` D ) )
11	5, 8	posref 16274	. . 3 |- ( ( O e. Poset /\ a e. ( Base ` O ) ) -> a ( le ` O ) a )
12		vex 3034	. . . 4 |- a e. _V
13	12, 12	brcnv 5022	. . 3 |- ( a `' ( le ` O ) a <-> a ( le ` O ) a )
14	11, 13	sylibr 217	. 2 |- ( ( O e. Poset /\ a e. ( Base ` O ) ) -> a `' ( le ` O ) a )
15		vex 3034	. . . . 5 |- b e. _V
16	12, 15	brcnv 5022	. . . 4 |- ( a `' ( le ` O ) b <-> b ( le ` O ) a )
17	15, 12	brcnv 5022	. . . 4 |- ( b `' ( le ` O ) a <-> a ( le ` O ) b )
18	16, 17	anbi12ci 712	. . 3 |- ( ( a `' ( le ` O ) b /\ b `' ( le ` O ) a ) <-> ( a ( le ` O ) b /\ b ( le ` O ) a ) )
19	5, 8	posasymb 16276	. . . 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 212	. . 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 225	. 2 |- ( ( O e. Poset /\ a e. ( Base ` O ) /\ b e. ( Base ` O ) ) -> ( ( a `' ( le ` O ) b /\ b `' ( le ` O ) a ) -> a = b ) )
22		3anrev 1018	. . . 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 16277	. . . 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 483	. . 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 3034	. . . . 5 |- c e. _V
26	15, 25	brcnv 5022	. . . 4 |- ( b `' ( le ` O ) c <-> c ( le ` O ) b )
27	16, 26	anbi12ci 712	. . 3 |- ( ( a `' ( le ` O ) b /\ b `' ( le ` O ) c ) <-> ( c ( le ` O ) b /\ b ( le ` O ) a ) )
28	12, 25	brcnv 5022	. . 3 |- ( a `' ( le ` O ) c <-> c ( le ` O ) a )
29	24, 27, 28	3imtr4g 278	. 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 16279	1 |- ( O e. Poset -> D e. Poset )

Syntax hints:   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   _Vcvv 3031   class class class wbr 4395   `'ccnv 4838   ` cfv 5589   Basecbs 15199   lecple 15275   Posetcpo 16263  ODualcodu 16452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ple 15288  df-preset 16251  df-poset 16269  df-odu 16453

This theorem is referenced by:  oduposb  16460  posglbd  16474  odutos  28499