Theorem odupos 15964

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 5858	. . . 4 |- (ODual ` O ) e. _V
3	1, 2	eqeltri 2538	. . 3 |- D e. _V
4	3	a1i 11	. 2 |- ( O e. Poset -> D e. _V )
5		eqid 2454	. . . 4 |- ( Base ` O ) = ( Base ` O )
6	1, 5	odubas 15962	. . 3 |- ( Base ` O ) = ( Base ` D )
7	6	a1i 11	. 2 |- ( O e. Poset -> ( Base ` O ) = ( Base ` D ) )
8		eqid 2454	. . . 4 |- ( le ` O ) = ( le ` O )
9	1, 8	oduleval 15960	. . 3 |- `' ( le ` O ) = ( le ` D )
10	9	a1i 11	. 2 |- ( O e. Poset -> `' ( le ` O ) = ( le ` D ) )
11	5, 8	posref 15779	. . 3 |- ( ( O e. Poset /\ a e. ( Base ` O ) ) -> a ( le ` O ) a )
12		vex 3109	. . . 4 |- a e. _V
13	12, 12	brcnv 5174	. . 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 3109	. . . . 5 |- b e. _V
16	12, 15	brcnv 5174	. . . 4 |- ( a `' ( le ` O ) b <-> b ( le ` O ) a )
17	15, 12	brcnv 5174	. . . 4 |- ( b `' ( le ` O ) a <-> a ( le ` O ) b )
18	16, 17	anbi12ci 696	. . 3 |- ( ( a `' ( le ` O ) b /\ b `' ( le ` O ) a ) <-> ( a ( le ` O ) b /\ b ( le ` O ) a ) )
19	5, 8	posasymb 15781	. . . 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 982	. . . 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 15782	. . . 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 473	. . 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 3109	. . . . 5 |- c e. _V
26	15, 25	brcnv 5174	. . . 4 |- ( b `' ( le ` O ) c <-> c ( le ` O ) b )
27	16, 26	anbi12ci 696	. . 3 |- ( ( a `' ( le ` O ) b /\ b `' ( le ` O ) c ) <-> ( c ( le ` O ) b /\ b ( le ` O ) a ) )
28	12, 25	brcnv 5174	. . 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 15784	1 |- ( O e. Poset -> D e. Poset )

Syntax hints:   -> wi 4   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   _Vcvv 3106   class class class wbr 4439   `'ccnv 4987   ` cfv 5570   Basecbs 14716   lecple 14791   Posetcpo 15768  ODualcodu 15957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ple 14804  df-preset 15756  df-poset 15774  df-odu 15958

This theorem is referenced by:  oduposb  15965  posglbd  15979  odutos  27885