Theorem odulatb 15972

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

Step	Hyp	Ref	Expression
1		oduglb.d	. . . 4 |- D = (ODual ` O )
2	1	oduposb 15965	. . 3 |- ( O e. V -> ( O e. Poset <-> D e. Poset ) )
3		ancom 448	. . . 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 ) ) ) )
4	3	a1i 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 ) ) ) ) )
5	2, 4	anbi12d 708	. 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 2454	. . 3 |- ( Base ` O ) = ( Base ` O )
7		eqid 2454	. . 3 |- ( join ` O ) = ( join ` O )
8		eqid 2454	. . 3 |- ( meet ` O ) = ( meet ` O )
9	6, 7, 8	islat 15876	. 2 |- ( O e. Lat <-> ( O e. Poset /\ ( dom ( join ` O ) = ( ( Base ` O ) X. ( Base ` O ) ) /\ dom ( meet ` O ) = ( ( Base ` O ) X. ( Base ` O ) ) ) ) )
10	1, 6	odubas 15962	. . 3 |- ( Base ` O ) = ( Base ` D )
11	1, 8	odujoin 15971	. . 3 |- ( meet ` O ) = ( join ` D )
12	1, 7	odumeet 15969	. . 3 |- ( join ` O ) = ( meet ` D )
13	10, 11, 12	islat 15876	. 2 |- ( D e. Lat <-> ( D e. Poset /\ ( dom ( meet ` O ) = ( ( Base ` O ) X. ( Base ` O ) ) /\ dom ( join ` O ) = ( ( Base ` O ) X. ( Base ` O ) ) ) ) )
14	5, 9, 13	3bitr4g 288	1 |- ( O e. V -> ( O e. Lat <-> D e. Lat ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   X. cxp 4986   dom cdm 4988   ` cfv 5570   Basecbs 14716   Posetcpo 15768   joincjn 15772   meetcmee 15773   Latclat 15874  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-rep 4550  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-rmo 2812  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-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-lat 15875  df-odu 15958

This theorem is referenced by:  odulat  15974  odudlatb  16025