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Theorem odudlatb 17019
 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 (𝐾𝑉 → (𝐾 ∈ DLat ↔ 𝐷 ∈ DLat))

Proof of Theorem odudlatb
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2610 . . . . . 6 (join‘𝐾) = (join‘𝐾)
3 eqid 2610 . . . . . 6 (meet‘𝐾) = (meet‘𝐾)
41, 2, 3latdisd 17013 . . . . 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))))
54bicomd 212 . . . 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))))
65pm5.32i 667 . . 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)
87odulatb 16966 . . . 4 (𝐾𝑉 → (𝐾 ∈ Lat ↔ 𝐷 ∈ Lat))
98anbi1d 737 . . 3 (𝐾𝑉 → ((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)))))
106, 9syl5bb 271 . 2 (𝐾𝑉 → ((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)))))
111, 2, 3isdlat 17016 . 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))))
127, 1odubas 16956 . . 3 (Base‘𝐾) = (Base‘𝐷)
137, 3odujoin 16965 . . 3 (meet‘𝐾) = (join‘𝐷)
147, 2odumeet 16963 . . 3 (join‘𝐾) = (meet‘𝐷)
1512, 13, 14isdlat 17016 . 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))))
1610, 11, 153bitr4g 302 1 (𝐾𝑉 → (𝐾 ∈ DLat ↔ 𝐷 ∈ DLat))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  joincjn 16767  meetcmee 16768  Latclat 16868  ODualcodu 16951  DLatcdlat 17014 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-dec 11370  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ple 15788  df-preset 16751  df-poset 16769  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-lat 16869  df-odu 16952  df-dlat 17015 This theorem is referenced by:  dlatjmdi  17020
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