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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 3515 | . 2 ⊢ (𝑂 ∈ CLat → 𝑂 ∈ V) | |
2 | noel 4299 | . . . . 5 ⊢ ¬ ((lub‘∅)‘∅) ∈ ∅ | |
3 | ssid 3992 | . . . . . 6 ⊢ ∅ ⊆ ∅ | |
4 | base0 16539 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
5 | eqid 2824 | . . . . . . 7 ⊢ (lub‘∅) = (lub‘∅) | |
6 | 4, 5 | clatlubcl 17725 | . . . . . 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 6666 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
11 | 9, 10 | syl5eq 2871 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → 𝐷 = ∅) |
12 | 11 | eleq1d 2900 | . . . 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 17749 | . . . 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 2824 | . . . . . . . . 9 ⊢ (glb‘𝑂) = (glb‘𝑂) | |
18 | 9, 17 | odulub 17754 | . . . . . . . 8 ⊢ (𝑂 ∈ V → (glb‘𝑂) = (lub‘𝐷)) |
19 | 18 | dmeqd 5777 | . . . . . . 7 ⊢ (𝑂 ∈ V → dom (glb‘𝑂) = dom (lub‘𝐷)) |
20 | 19 | eqeq1d 2826 | . . . . . 6 ⊢ (𝑂 ∈ V → (dom (glb‘𝑂) = 𝒫 (Base‘𝑂) ↔ dom (lub‘𝐷) = 𝒫 (Base‘𝑂))) |
21 | eqid 2824 | . . . . . . . . 9 ⊢ (lub‘𝑂) = (lub‘𝑂) | |
22 | 9, 21 | oduglb 17752 | . . . . . . . 8 ⊢ (𝑂 ∈ V → (lub‘𝑂) = (glb‘𝐷)) |
23 | 22 | dmeqd 5777 | . . . . . . 7 ⊢ (𝑂 ∈ V → dom (lub‘𝑂) = dom (glb‘𝐷)) |
24 | 23 | eqeq1d 2826 | . . . . . 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 2824 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
29 | 28, 21, 17 | isclat 17722 | . . 3 ⊢ (𝑂 ∈ CLat ↔ (𝑂 ∈ Poset ∧ (dom (lub‘𝑂) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝑂) = 𝒫 (Base‘𝑂)))) |
30 | 9, 28 | odubas 17746 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝐷) |
31 | eqid 2824 | . . . 4 ⊢ (lub‘𝐷) = (lub‘𝐷) | |
32 | eqid 2824 | . . . 4 ⊢ (glb‘𝐷) = (glb‘𝐷) | |
33 | 30, 31, 32 | isclat 17722 | . . 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 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 ∅c0 4294 𝒫 cpw 4542 dom cdm 5558 ‘cfv 6358 Basecbs 16486 Posetcpo 17553 lubclub 17555 glbcglb 17556 CLatccla 17720 ODualcodu 17741 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-dec 12102 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ple 16588 df-proset 17541 df-poset 17559 df-lub 17587 df-glb 17588 df-clat 17721 df-odu 17742 |
This theorem is referenced by: (None) |
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