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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 3501 | . 2 ⊢ (𝑂 ∈ CLat → 𝑂 ∈ V) | |
| 2 | noel 4338 | . . . . 5 ⊢ ¬ ((lub‘∅)‘∅) ∈ ∅ | |
| 3 | ssid 4006 | . . . . . 6 ⊢ ∅ ⊆ ∅ | |
| 4 | base0 17252 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
| 5 | eqid 2737 | . . . . . . 7 ⊢ (lub‘∅) = (lub‘∅) | |
| 6 | 4, 5 | clatlubcl 18548 | . . . . . 6 ⊢ ((∅ ∈ CLat ∧ ∅ ⊆ ∅) → ((lub‘∅)‘∅) ∈ ∅) |
| 7 | 3, 6 | mpan2 691 | . . . . 5 ⊢ (∅ ∈ CLat → ((lub‘∅)‘∅) ∈ ∅) |
| 8 | 2, 7 | mto 197 | . . . 4 ⊢ ¬ ∅ ∈ CLat |
| 9 | oduclatb.d | . . . . . 6 ⊢ 𝐷 = (ODual‘𝑂) | |
| 10 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
| 11 | 9, 10 | eqtrid 2789 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → 𝐷 = ∅) |
| 12 | 11 | eleq1d 2826 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (𝐷 ∈ CLat ↔ ∅ ∈ CLat)) |
| 13 | 8, 12 | mtbiri 327 | . . 3 ⊢ (¬ 𝑂 ∈ V → ¬ 𝐷 ∈ CLat) |
| 14 | 13 | con4i 114 | . 2 ⊢ (𝐷 ∈ CLat → 𝑂 ∈ V) |
| 15 | 9 | oduposb 18374 | . . . 4 ⊢ (𝑂 ∈ V → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) |
| 16 | ancom 460 | . . . . 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 2737 | . . . . . . . . 9 ⊢ (glb‘𝑂) = (glb‘𝑂) | |
| 18 | 9, 17 | odulub 18452 | . . . . . . . 8 ⊢ (𝑂 ∈ V → (glb‘𝑂) = (lub‘𝐷)) |
| 19 | 18 | dmeqd 5916 | . . . . . . 7 ⊢ (𝑂 ∈ V → dom (glb‘𝑂) = dom (lub‘𝐷)) |
| 20 | 19 | eqeq1d 2739 | . . . . . 6 ⊢ (𝑂 ∈ V → (dom (glb‘𝑂) = 𝒫 (Base‘𝑂) ↔ dom (lub‘𝐷) = 𝒫 (Base‘𝑂))) |
| 21 | eqid 2737 | . . . . . . . . 9 ⊢ (lub‘𝑂) = (lub‘𝑂) | |
| 22 | 9, 21 | oduglb 18454 | . . . . . . . 8 ⊢ (𝑂 ∈ V → (lub‘𝑂) = (glb‘𝐷)) |
| 23 | 22 | dmeqd 5916 | . . . . . . 7 ⊢ (𝑂 ∈ V → dom (lub‘𝑂) = dom (glb‘𝐷)) |
| 24 | 23 | eqeq1d 2739 | . . . . . 6 ⊢ (𝑂 ∈ V → (dom (lub‘𝑂) = 𝒫 (Base‘𝑂) ↔ dom (glb‘𝐷) = 𝒫 (Base‘𝑂))) |
| 25 | 20, 24 | anbi12d 632 | . . . . 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 283 | . . . 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 632 | . . 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 2737 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
| 29 | 28, 21, 17 | isclat 18545 | . . 3 ⊢ (𝑂 ∈ CLat ↔ (𝑂 ∈ Poset ∧ (dom (lub‘𝑂) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝑂) = 𝒫 (Base‘𝑂)))) |
| 30 | 9, 28 | odubas 18336 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝐷) |
| 31 | eqid 2737 | . . . 4 ⊢ (lub‘𝐷) = (lub‘𝐷) | |
| 32 | eqid 2737 | . . . 4 ⊢ (glb‘𝐷) = (glb‘𝐷) | |
| 33 | 30, 31, 32 | isclat 18545 | . . 3 ⊢ (𝐷 ∈ CLat ↔ (𝐷 ∈ Poset ∧ (dom (lub‘𝐷) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝐷) = 𝒫 (Base‘𝑂)))) |
| 34 | 27, 29, 33 | 3bitr4g 314 | . 2 ⊢ (𝑂 ∈ V → (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat)) |
| 35 | 1, 14, 34 | pm5.21nii 378 | 1 ⊢ (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 dom cdm 5685 ‘cfv 6561 Basecbs 17247 ODualcodu 18331 Posetcpo 18353 lubclub 18355 glbcglb 18356 CLatccla 18543 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-dec 12734 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ple 17317 df-odu 18332 df-proset 18340 df-poset 18359 df-lub 18391 df-glb 18392 df-clat 18544 |
| This theorem is referenced by: (None) |
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