Theorem odudlatb 18595

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 2740	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2740	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2740	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
4	1, 2, 3	latdisd 18567	. . . . 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 18504	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Lat ↔ 𝐷 ∈ Lat))
9	8	anbi1d 630	. . 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 18592	. 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 18361	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐷)
13	7, 3	odujoin 18478	. . 3 ⊢ (meet‘𝐾) = (join‘𝐷)
14	7, 2	odumeet 18480	. . 3 ⊢ (join‘𝐾) = (meet‘𝐷)
15	12, 13, 14	isdlat 18592	. 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 1537   ∈ wcel 2108  ∀wral 3067  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  ODualcodu 18356  joincjn 18381  meetcmee 18382  Latclat 18501  DLatcdlat 18590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-dec 12759  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ple 17331  df-odu 18357  df-proset 18365  df-poset 18383  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-lat 18502  df-dlat 18591

This theorem is referenced by:  dlatjmdi  18596