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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 |
---|---|
oduglb.d | ⊢ 𝐷 = (ODual‘𝑂) |
Ref | Expression |
---|---|
oduclatb | ⊢ (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3426 | . 2 ⊢ (𝑂 ∈ CLat → 𝑂 ∈ V) | |
2 | noel 4177 | . . . . 5 ⊢ ¬ ((lub‘∅)‘∅) ∈ ∅ | |
3 | ssid 3872 | . . . . . 6 ⊢ ∅ ⊆ ∅ | |
4 | base0 16390 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
5 | eqid 2771 | . . . . . . 7 ⊢ (lub‘∅) = (lub‘∅) | |
6 | 4, 5 | clatlubcl 17592 | . . . . . 6 ⊢ ((∅ ∈ CLat ∧ ∅ ⊆ ∅) → ((lub‘∅)‘∅) ∈ ∅) |
7 | 3, 6 | mpan2 679 | . . . . 5 ⊢ (∅ ∈ CLat → ((lub‘∅)‘∅) ∈ ∅) |
8 | 2, 7 | mto 189 | . . . 4 ⊢ ¬ ∅ ∈ CLat |
9 | oduglb.d | . . . . . 6 ⊢ 𝐷 = (ODual‘𝑂) | |
10 | fvprc 6489 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
11 | 9, 10 | syl5eq 2819 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → 𝐷 = ∅) |
12 | 11 | eleq1d 2843 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (𝐷 ∈ CLat ↔ ∅ ∈ CLat)) |
13 | 8, 12 | mtbiri 319 | . . 3 ⊢ (¬ 𝑂 ∈ V → ¬ 𝐷 ∈ CLat) |
14 | 13 | con4i 114 | . 2 ⊢ (𝐷 ∈ CLat → 𝑂 ∈ V) |
15 | 9 | oduposb 17616 | . . . 4 ⊢ (𝑂 ∈ V → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) |
16 | ancom 453 | . . . . 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 2771 | . . . . . . . . 9 ⊢ (glb‘𝑂) = (glb‘𝑂) | |
18 | 9, 17 | odulub 17621 | . . . . . . . 8 ⊢ (𝑂 ∈ V → (glb‘𝑂) = (lub‘𝐷)) |
19 | 18 | dmeqd 5620 | . . . . . . 7 ⊢ (𝑂 ∈ V → dom (glb‘𝑂) = dom (lub‘𝐷)) |
20 | 19 | eqeq1d 2773 | . . . . . 6 ⊢ (𝑂 ∈ V → (dom (glb‘𝑂) = 𝒫 (Base‘𝑂) ↔ dom (lub‘𝐷) = 𝒫 (Base‘𝑂))) |
21 | eqid 2771 | . . . . . . . . 9 ⊢ (lub‘𝑂) = (lub‘𝑂) | |
22 | 9, 21 | oduglb 17619 | . . . . . . . 8 ⊢ (𝑂 ∈ V → (lub‘𝑂) = (glb‘𝐷)) |
23 | 22 | dmeqd 5620 | . . . . . . 7 ⊢ (𝑂 ∈ V → dom (lub‘𝑂) = dom (glb‘𝐷)) |
24 | 23 | eqeq1d 2773 | . . . . . 6 ⊢ (𝑂 ∈ V → (dom (lub‘𝑂) = 𝒫 (Base‘𝑂) ↔ dom (glb‘𝐷) = 𝒫 (Base‘𝑂))) |
25 | 20, 24 | anbi12d 622 | . . . . 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 | syl5bb 275 | . . . 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 622 | . . 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 2771 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
29 | 28, 21, 17 | isclat 17589 | . . 3 ⊢ (𝑂 ∈ CLat ↔ (𝑂 ∈ Poset ∧ (dom (lub‘𝑂) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝑂) = 𝒫 (Base‘𝑂)))) |
30 | 9, 28 | odubas 17613 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝐷) |
31 | eqid 2771 | . . . 4 ⊢ (lub‘𝐷) = (lub‘𝐷) | |
32 | eqid 2771 | . . . 4 ⊢ (glb‘𝐷) = (glb‘𝐷) | |
33 | 30, 31, 32 | isclat 17589 | . . 3 ⊢ (𝐷 ∈ CLat ↔ (𝐷 ∈ Poset ∧ (dom (lub‘𝐷) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝐷) = 𝒫 (Base‘𝑂)))) |
34 | 27, 29, 33 | 3bitr4g 306 | . 2 ⊢ (𝑂 ∈ V → (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat)) |
35 | 1, 14, 34 | pm5.21nii 371 | 1 ⊢ (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 Vcvv 3408 ⊆ wss 3822 ∅c0 4172 𝒫 cpw 4416 dom cdm 5403 ‘cfv 6185 Basecbs 16337 Posetcpo 17420 lubclub 17422 glbcglb 17423 CLatccla 17587 ODualcodu 17608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-dec 11910 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ple 16439 df-proset 17408 df-poset 17426 df-lub 17454 df-glb 17455 df-clat 17588 df-odu 17609 |
This theorem is referenced by: (None) |
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