Theorem odudlatb 18431

Description: The dual of a distributive lattice is a distributive lattice and conversely. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Hypothesis

Ref	Expression
odudlat.d	⊢ 𝐷 = (ODual‘𝐾)

Assertion

Ref	Expression
odudlatb	⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ DLat ↔ 𝐷 ∈ DLat))

Proof of Theorem odudlatb

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2729	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2729	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
4	1, 2, 3	latdisd 18403	. . . . 5 ⊢ (𝐾 ∈ Lat → (∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(join‘𝐾)(𝑦(meet‘𝐾)𝑧)) = ((𝑥(join‘𝐾)𝑦)(meet‘𝐾)(𝑥(join‘𝐾)𝑧)) ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(meet‘𝐾)(𝑦(join‘𝐾)𝑧)) = ((𝑥(meet‘𝐾)𝑦)(join‘𝐾)(𝑥(meet‘𝐾)𝑧))))
5	4	bicomd 223	. . . 4 ⊢ (𝐾 ∈ Lat → (∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(meet‘𝐾)(𝑦(join‘𝐾)𝑧)) = ((𝑥(meet‘𝐾)𝑦)(join‘𝐾)(𝑥(meet‘𝐾)𝑧)) ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(join‘𝐾)(𝑦(meet‘𝐾)𝑧)) = ((𝑥(join‘𝐾)𝑦)(meet‘𝐾)(𝑥(join‘𝐾)𝑧))))
6	5	pm5.32i 574	. . 3 ⊢ ((𝐾 ∈ Lat ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(meet‘𝐾)(𝑦(join‘𝐾)𝑧)) = ((𝑥(meet‘𝐾)𝑦)(join‘𝐾)(𝑥(meet‘𝐾)𝑧))) ↔ (𝐾 ∈ Lat ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(join‘𝐾)(𝑦(meet‘𝐾)𝑧)) = ((𝑥(join‘𝐾)𝑦)(meet‘𝐾)(𝑥(join‘𝐾)𝑧))))
7		odudlat.d	. . . . 5 ⊢ 𝐷 = (ODual‘𝐾)
8	7	odulatb 18340	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Lat ↔ 𝐷 ∈ Lat))
9	8	anbi1d 631	. . 3 ⊢ (𝐾 ∈ 𝑉 → ((𝐾 ∈ Lat ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(join‘𝐾)(𝑦(meet‘𝐾)𝑧)) = ((𝑥(join‘𝐾)𝑦)(meet‘𝐾)(𝑥(join‘𝐾)𝑧))) ↔ (𝐷 ∈ Lat ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(join‘𝐾)(𝑦(meet‘𝐾)𝑧)) = ((𝑥(join‘𝐾)𝑦)(meet‘𝐾)(𝑥(join‘𝐾)𝑧)))))
10	6, 9	bitrid 283	. 2 ⊢ (𝐾 ∈ 𝑉 → ((𝐾 ∈ Lat ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(meet‘𝐾)(𝑦(join‘𝐾)𝑧)) = ((𝑥(meet‘𝐾)𝑦)(join‘𝐾)(𝑥(meet‘𝐾)𝑧))) ↔ (𝐷 ∈ Lat ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(join‘𝐾)(𝑦(meet‘𝐾)𝑧)) = ((𝑥(join‘𝐾)𝑦)(meet‘𝐾)(𝑥(join‘𝐾)𝑧)))))
11	1, 2, 3	isdlat 18428	. 2 ⊢ (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(meet‘𝐾)(𝑦(join‘𝐾)𝑧)) = ((𝑥(meet‘𝐾)𝑦)(join‘𝐾)(𝑥(meet‘𝐾)𝑧))))
12	7, 1	odubas 18197	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐷)
13	7, 3	odujoin 18312	. . 3 ⊢ (meet‘𝐾) = (join‘𝐷)
14	7, 2	odumeet 18314	. . 3 ⊢ (join‘𝐾) = (meet‘𝐷)
15	12, 13, 14	isdlat 18428	. 2 ⊢ (𝐷 ∈ DLat ↔ (𝐷 ∈ Lat ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(join‘𝐾)(𝑦(meet‘𝐾)𝑧)) = ((𝑥(join‘𝐾)𝑦)(meet‘𝐾)(𝑥(join‘𝐾)𝑧))))
16	10, 11, 15	3bitr4g 314	1 ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ DLat ↔ 𝐷 ∈ DLat))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  ODualcodu 18192  joincjn 18217  meetcmee 18218  Latclat 18337  DLatcdlat 18426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-dec 12592  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ple 17181  df-odu 18193  df-proset 18200  df-poset 18219  df-lub 18250  df-glb 18251  df-join 18252  df-meet 18253  df-lat 18338  df-dlat 18427

This theorem is referenced by:  dlatjmdi  18432