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Mirrors > Home > MPE Home > Th. List > oduclatb | Structured version Visualization version GIF version |
Description: Being a complete lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
Ref | Expression |
---|---|
oduclatb.d | ⊢ 𝐷 = (ODual‘𝑂) |
Ref | Expression |
---|---|
oduclatb | ⊢ (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3482 | . 2 ⊢ (𝑂 ∈ CLat → 𝑂 ∈ V) | |
2 | noel 4333 | . . . . 5 ⊢ ¬ ((lub‘∅)‘∅) ∈ ∅ | |
3 | ssid 4002 | . . . . . 6 ⊢ ∅ ⊆ ∅ | |
4 | base0 17218 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
5 | eqid 2726 | . . . . . . 7 ⊢ (lub‘∅) = (lub‘∅) | |
6 | 4, 5 | clatlubcl 18528 | . . . . . 6 ⊢ ((∅ ∈ CLat ∧ ∅ ⊆ ∅) → ((lub‘∅)‘∅) ∈ ∅) |
7 | 3, 6 | mpan2 689 | . . . . 5 ⊢ (∅ ∈ CLat → ((lub‘∅)‘∅) ∈ ∅) |
8 | 2, 7 | mto 196 | . . . 4 ⊢ ¬ ∅ ∈ CLat |
9 | oduclatb.d | . . . . . 6 ⊢ 𝐷 = (ODual‘𝑂) | |
10 | fvprc 6893 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
11 | 9, 10 | eqtrid 2778 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → 𝐷 = ∅) |
12 | 11 | eleq1d 2811 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (𝐷 ∈ CLat ↔ ∅ ∈ CLat)) |
13 | 8, 12 | mtbiri 326 | . . 3 ⊢ (¬ 𝑂 ∈ V → ¬ 𝐷 ∈ CLat) |
14 | 13 | con4i 114 | . 2 ⊢ (𝐷 ∈ CLat → 𝑂 ∈ V) |
15 | 9 | oduposb 18354 | . . . 4 ⊢ (𝑂 ∈ V → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) |
16 | ancom 459 | . . . . 5 ⊢ ((dom (lub‘𝑂) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝑂) = 𝒫 (Base‘𝑂)) ↔ (dom (glb‘𝑂) = 𝒫 (Base‘𝑂) ∧ dom (lub‘𝑂) = 𝒫 (Base‘𝑂))) | |
17 | eqid 2726 | . . . . . . . . 9 ⊢ (glb‘𝑂) = (glb‘𝑂) | |
18 | 9, 17 | odulub 18432 | . . . . . . . 8 ⊢ (𝑂 ∈ V → (glb‘𝑂) = (lub‘𝐷)) |
19 | 18 | dmeqd 5912 | . . . . . . 7 ⊢ (𝑂 ∈ V → dom (glb‘𝑂) = dom (lub‘𝐷)) |
20 | 19 | eqeq1d 2728 | . . . . . 6 ⊢ (𝑂 ∈ V → (dom (glb‘𝑂) = 𝒫 (Base‘𝑂) ↔ dom (lub‘𝐷) = 𝒫 (Base‘𝑂))) |
21 | eqid 2726 | . . . . . . . . 9 ⊢ (lub‘𝑂) = (lub‘𝑂) | |
22 | 9, 21 | oduglb 18434 | . . . . . . . 8 ⊢ (𝑂 ∈ V → (lub‘𝑂) = (glb‘𝐷)) |
23 | 22 | dmeqd 5912 | . . . . . . 7 ⊢ (𝑂 ∈ V → dom (lub‘𝑂) = dom (glb‘𝐷)) |
24 | 23 | eqeq1d 2728 | . . . . . 6 ⊢ (𝑂 ∈ V → (dom (lub‘𝑂) = 𝒫 (Base‘𝑂) ↔ dom (glb‘𝐷) = 𝒫 (Base‘𝑂))) |
25 | 20, 24 | anbi12d 630 | . . . . 5 ⊢ (𝑂 ∈ V → ((dom (glb‘𝑂) = 𝒫 (Base‘𝑂) ∧ dom (lub‘𝑂) = 𝒫 (Base‘𝑂)) ↔ (dom (lub‘𝐷) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝐷) = 𝒫 (Base‘𝑂)))) |
26 | 16, 25 | bitrid 282 | . . . 4 ⊢ (𝑂 ∈ V → ((dom (lub‘𝑂) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝑂) = 𝒫 (Base‘𝑂)) ↔ (dom (lub‘𝐷) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝐷) = 𝒫 (Base‘𝑂)))) |
27 | 15, 26 | anbi12d 630 | . . 3 ⊢ (𝑂 ∈ V → ((𝑂 ∈ Poset ∧ (dom (lub‘𝑂) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝑂) = 𝒫 (Base‘𝑂))) ↔ (𝐷 ∈ Poset ∧ (dom (lub‘𝐷) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝐷) = 𝒫 (Base‘𝑂))))) |
28 | eqid 2726 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
29 | 28, 21, 17 | isclat 18525 | . . 3 ⊢ (𝑂 ∈ CLat ↔ (𝑂 ∈ Poset ∧ (dom (lub‘𝑂) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝑂) = 𝒫 (Base‘𝑂)))) |
30 | 9, 28 | odubas 18316 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝐷) |
31 | eqid 2726 | . . . 4 ⊢ (lub‘𝐷) = (lub‘𝐷) | |
32 | eqid 2726 | . . . 4 ⊢ (glb‘𝐷) = (glb‘𝐷) | |
33 | 30, 31, 32 | isclat 18525 | . . 3 ⊢ (𝐷 ∈ CLat ↔ (𝐷 ∈ Poset ∧ (dom (lub‘𝐷) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝐷) = 𝒫 (Base‘𝑂)))) |
34 | 27, 29, 33 | 3bitr4g 313 | . 2 ⊢ (𝑂 ∈ V → (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat)) |
35 | 1, 14, 34 | pm5.21nii 377 | 1 ⊢ (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3947 ∅c0 4325 𝒫 cpw 4607 dom cdm 5682 ‘cfv 6554 Basecbs 17213 ODualcodu 18311 Posetcpo 18332 lubclub 18334 glbcglb 18335 CLatccla 18523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-dec 12730 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ple 17286 df-odu 18312 df-proset 18320 df-poset 18338 df-lub 18371 df-glb 18372 df-clat 18524 |
This theorem is referenced by: (None) |
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