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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 3513 | . 2 ⊢ (𝑂 ∈ CLat → 𝑂 ∈ V) | |
2 | noel 4296 | . . . . 5 ⊢ ¬ ((lub‘∅)‘∅) ∈ ∅ | |
3 | ssid 3989 | . . . . . 6 ⊢ ∅ ⊆ ∅ | |
4 | base0 16530 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
5 | eqid 2821 | . . . . . . 7 ⊢ (lub‘∅) = (lub‘∅) | |
6 | 4, 5 | clatlubcl 17716 | . . . . . 6 ⊢ ((∅ ∈ CLat ∧ ∅ ⊆ ∅) → ((lub‘∅)‘∅) ∈ ∅) |
7 | 3, 6 | mpan2 689 | . . . . 5 ⊢ (∅ ∈ CLat → ((lub‘∅)‘∅) ∈ ∅) |
8 | 2, 7 | mto 199 | . . . 4 ⊢ ¬ ∅ ∈ CLat |
9 | oduglb.d | . . . . . 6 ⊢ 𝐷 = (ODual‘𝑂) | |
10 | fvprc 6658 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
11 | 9, 10 | syl5eq 2868 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → 𝐷 = ∅) |
12 | 11 | eleq1d 2897 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (𝐷 ∈ CLat ↔ ∅ ∈ CLat)) |
13 | 8, 12 | mtbiri 329 | . . 3 ⊢ (¬ 𝑂 ∈ V → ¬ 𝐷 ∈ CLat) |
14 | 13 | con4i 114 | . 2 ⊢ (𝐷 ∈ CLat → 𝑂 ∈ V) |
15 | 9 | oduposb 17740 | . . . 4 ⊢ (𝑂 ∈ V → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) |
16 | ancom 463 | . . . . 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 2821 | . . . . . . . . 9 ⊢ (glb‘𝑂) = (glb‘𝑂) | |
18 | 9, 17 | odulub 17745 | . . . . . . . 8 ⊢ (𝑂 ∈ V → (glb‘𝑂) = (lub‘𝐷)) |
19 | 18 | dmeqd 5769 | . . . . . . 7 ⊢ (𝑂 ∈ V → dom (glb‘𝑂) = dom (lub‘𝐷)) |
20 | 19 | eqeq1d 2823 | . . . . . 6 ⊢ (𝑂 ∈ V → (dom (glb‘𝑂) = 𝒫 (Base‘𝑂) ↔ dom (lub‘𝐷) = 𝒫 (Base‘𝑂))) |
21 | eqid 2821 | . . . . . . . . 9 ⊢ (lub‘𝑂) = (lub‘𝑂) | |
22 | 9, 21 | oduglb 17743 | . . . . . . . 8 ⊢ (𝑂 ∈ V → (lub‘𝑂) = (glb‘𝐷)) |
23 | 22 | dmeqd 5769 | . . . . . . 7 ⊢ (𝑂 ∈ V → dom (lub‘𝑂) = dom (glb‘𝐷)) |
24 | 23 | eqeq1d 2823 | . . . . . 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 | syl5bb 285 | . . . 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 2821 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
29 | 28, 21, 17 | isclat 17713 | . . 3 ⊢ (𝑂 ∈ CLat ↔ (𝑂 ∈ Poset ∧ (dom (lub‘𝑂) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝑂) = 𝒫 (Base‘𝑂)))) |
30 | 9, 28 | odubas 17737 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝐷) |
31 | eqid 2821 | . . . 4 ⊢ (lub‘𝐷) = (lub‘𝐷) | |
32 | eqid 2821 | . . . 4 ⊢ (glb‘𝐷) = (glb‘𝐷) | |
33 | 30, 31, 32 | isclat 17713 | . . 3 ⊢ (𝐷 ∈ CLat ↔ (𝐷 ∈ Poset ∧ (dom (lub‘𝐷) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝐷) = 𝒫 (Base‘𝑂)))) |
34 | 27, 29, 33 | 3bitr4g 316 | . 2 ⊢ (𝑂 ∈ V → (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat)) |
35 | 1, 14, 34 | pm5.21nii 382 | 1 ⊢ (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 dom cdm 5550 ‘cfv 6350 Basecbs 16477 Posetcpo 17544 lubclub 17546 glbcglb 17547 CLatccla 17711 ODualcodu 17732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-dec 12093 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ple 16579 df-proset 17532 df-poset 17550 df-lub 17578 df-glb 17579 df-clat 17712 df-odu 17733 |
This theorem is referenced by: (None) |
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