Theorem odulatb 15417

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 15410	. . 3 |- ( O e. V -> ( O e. Poset <-> D e. Poset ) )
3		ancom 450	. . . 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 710	. 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 2451	. . 3 |- ( Base ` O ) = ( Base ` O )
7		eqid 2451	. . 3 |- ( join ` O ) = ( join ` O )
8		eqid 2451	. . 3 |- ( meet ` O ) = ( meet ` O )
9	6, 7, 8	islat 15321	. 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 15407	. . 3 |- ( Base ` O ) = ( Base ` D )
11	1, 8	odujoin 15416	. . 3 |- ( meet ` O ) = ( join ` D )
12	1, 7	odumeet 15414	. . 3 |- ( join ` O ) = ( meet ` D )
13	10, 11, 12	islat 15321	. 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 369   = wceq 1370   e. wcel 1758   X. cxp 4938   dom cdm 4940   ` cfv 5518   Basecbs 14278   Posetcpo 15214   joincjn 15218   meetcmee 15219   Latclat 15319  ODualcodu 15402

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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462

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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ple 14362  df-poset 15220  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-lat 15320  df-odu 15403

This theorem is referenced by:  odulat  15419  odudlatb  15470