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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 3185 | . 2 ⊢ (𝑂 ∈ CLat → 𝑂 ∈ V) | |
2 | noel 3878 | . . . . 5 ⊢ ¬ ((lub‘∅)‘∅) ∈ ∅ | |
3 | ssid 3587 | . . . . . 6 ⊢ ∅ ⊆ ∅ | |
4 | base0 15740 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
5 | eqid 2610 | . . . . . . 7 ⊢ (lub‘∅) = (lub‘∅) | |
6 | 4, 5 | clatlubcl 16935 | . . . . . 6 ⊢ ((∅ ∈ CLat ∧ ∅ ⊆ ∅) → ((lub‘∅)‘∅) ∈ ∅) |
7 | 3, 6 | mpan2 703 | . . . . 5 ⊢ (∅ ∈ CLat → ((lub‘∅)‘∅) ∈ ∅) |
8 | 2, 7 | mto 187 | . . . 4 ⊢ ¬ ∅ ∈ CLat |
9 | oduglb.d | . . . . . 6 ⊢ 𝐷 = (ODual‘𝑂) | |
10 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅) | |
11 | 9, 10 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → 𝐷 = ∅) |
12 | 11 | eleq1d 2672 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (𝐷 ∈ CLat ↔ ∅ ∈ CLat)) |
13 | 8, 12 | mtbiri 316 | . . 3 ⊢ (¬ 𝑂 ∈ V → ¬ 𝐷 ∈ CLat) |
14 | 13 | con4i 112 | . 2 ⊢ (𝐷 ∈ CLat → 𝑂 ∈ V) |
15 | 9 | oduposb 16959 | . . . 4 ⊢ (𝑂 ∈ V → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) |
16 | ancom 465 | . . . . 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 2610 | . . . . . . . . 9 ⊢ (glb‘𝑂) = (glb‘𝑂) | |
18 | 9, 17 | odulub 16964 | . . . . . . . 8 ⊢ (𝑂 ∈ V → (glb‘𝑂) = (lub‘𝐷)) |
19 | 18 | dmeqd 5248 | . . . . . . 7 ⊢ (𝑂 ∈ V → dom (glb‘𝑂) = dom (lub‘𝐷)) |
20 | 19 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑂 ∈ V → (dom (glb‘𝑂) = 𝒫 (Base‘𝑂) ↔ dom (lub‘𝐷) = 𝒫 (Base‘𝑂))) |
21 | eqid 2610 | . . . . . . . . 9 ⊢ (lub‘𝑂) = (lub‘𝑂) | |
22 | 9, 21 | oduglb 16962 | . . . . . . . 8 ⊢ (𝑂 ∈ V → (lub‘𝑂) = (glb‘𝐷)) |
23 | 22 | dmeqd 5248 | . . . . . . 7 ⊢ (𝑂 ∈ V → dom (lub‘𝑂) = dom (glb‘𝐷)) |
24 | 23 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑂 ∈ V → (dom (lub‘𝑂) = 𝒫 (Base‘𝑂) ↔ dom (glb‘𝐷) = 𝒫 (Base‘𝑂))) |
25 | 20, 24 | anbi12d 743 | . . . . 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 271 | . . . 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 743 | . . 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 2610 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
29 | 28, 21, 17 | isclat 16932 | . . 3 ⊢ (𝑂 ∈ CLat ↔ (𝑂 ∈ Poset ∧ (dom (lub‘𝑂) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝑂) = 𝒫 (Base‘𝑂)))) |
30 | 9, 28 | odubas 16956 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝐷) |
31 | eqid 2610 | . . . 4 ⊢ (lub‘𝐷) = (lub‘𝐷) | |
32 | eqid 2610 | . . . 4 ⊢ (glb‘𝐷) = (glb‘𝐷) | |
33 | 30, 31, 32 | isclat 16932 | . . 3 ⊢ (𝐷 ∈ CLat ↔ (𝐷 ∈ Poset ∧ (dom (lub‘𝐷) = 𝒫 (Base‘𝑂) ∧ dom (glb‘𝐷) = 𝒫 (Base‘𝑂)))) |
34 | 27, 29, 33 | 3bitr4g 302 | . 2 ⊢ (𝑂 ∈ V → (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat)) |
35 | 1, 14, 34 | pm5.21nii 367 | 1 ⊢ (𝑂 ∈ CLat ↔ 𝐷 ∈ CLat) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 dom cdm 5038 ‘cfv 5804 Basecbs 15695 Posetcpo 16763 lubclub 16765 glbcglb 16766 CLatccla 16930 ODualcodu 16951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-dec 11370 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ple 15788 df-preset 16751 df-poset 16769 df-lub 16797 df-glb 16798 df-clat 16931 df-odu 16952 |
This theorem is referenced by: (None) |
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