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Mirrors > Home > MPE Home > Th. List > oduposb | Structured version Visualization version GIF version |
Description: Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
Ref | Expression |
---|---|
odupos.d | ⊢ 𝐷 = (ODual‘𝑂) |
Ref | Expression |
---|---|
oduposb | ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odupos.d | . . 3 ⊢ 𝐷 = (ODual‘𝑂) | |
2 | 1 | odupos 17847 | . 2 ⊢ (𝑂 ∈ Poset → 𝐷 ∈ Poset) |
3 | eqid 2738 | . . . 4 ⊢ (ODual‘𝐷) = (ODual‘𝐷) | |
4 | 3 | odupos 17847 | . . 3 ⊢ (𝐷 ∈ Poset → (ODual‘𝐷) ∈ Poset) |
5 | fvexd 6741 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (ODual‘𝐷) ∈ V) | |
6 | id 22 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝑉) | |
7 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
8 | 1, 7 | odubas 17812 | . . . . . 6 ⊢ (Base‘𝑂) = (Base‘𝐷) |
9 | 3, 8 | odubas 17812 | . . . . 5 ⊢ (Base‘𝑂) = (Base‘(ODual‘𝐷)) |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (Base‘𝑂) = (Base‘(ODual‘𝐷))) |
11 | eqidd 2739 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (Base‘𝑂) = (Base‘𝑂)) | |
12 | eqid 2738 | . . . . . . . . . 10 ⊢ (le‘𝑂) = (le‘𝑂) | |
13 | 1, 12 | oduleval 17810 | . . . . . . . . 9 ⊢ ◡(le‘𝑂) = (le‘𝐷) |
14 | 3, 13 | oduleval 17810 | . . . . . . . 8 ⊢ ◡◡(le‘𝑂) = (le‘(ODual‘𝐷)) |
15 | 14 | eqcomi 2747 | . . . . . . 7 ⊢ (le‘(ODual‘𝐷)) = ◡◡(le‘𝑂) |
16 | 15 | breqi 5068 | . . . . . 6 ⊢ (𝑎(le‘(ODual‘𝐷))𝑏 ↔ 𝑎◡◡(le‘𝑂)𝑏) |
17 | vex 3419 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
18 | vex 3419 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
19 | 17, 18 | brcnv 5760 | . . . . . 6 ⊢ (𝑎◡◡(le‘𝑂)𝑏 ↔ 𝑏◡(le‘𝑂)𝑎) |
20 | 18, 17 | brcnv 5760 | . . . . . 6 ⊢ (𝑏◡(le‘𝑂)𝑎 ↔ 𝑎(le‘𝑂)𝑏) |
21 | 16, 19, 20 | 3bitri 300 | . . . . 5 ⊢ (𝑎(le‘(ODual‘𝐷))𝑏 ↔ 𝑎(le‘𝑂)𝑏) |
22 | 21 | a1i 11 | . . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ (𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂))) → (𝑎(le‘(ODual‘𝐷))𝑏 ↔ 𝑎(le‘𝑂)𝑏)) |
23 | 5, 6, 10, 11, 22 | pospropd 17846 | . . 3 ⊢ (𝑂 ∈ 𝑉 → ((ODual‘𝐷) ∈ Poset ↔ 𝑂 ∈ Poset)) |
24 | 4, 23 | syl5ib 247 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝐷 ∈ Poset → 𝑂 ∈ Poset)) |
25 | 2, 24 | impbid2 229 | 1 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2111 Vcvv 3415 class class class wbr 5062 ◡ccnv 5559 ‘cfv 6389 Basecbs 16773 lecple 16822 ODualcodu 17807 Posetcpo 17827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 ax-cnex 10798 ax-resscn 10799 ax-1cn 10800 ax-icn 10801 ax-addcl 10802 ax-addrcl 10803 ax-mulcl 10804 ax-mulrcl 10805 ax-mulcom 10806 ax-addass 10807 ax-mulass 10808 ax-distr 10809 ax-i2m1 10810 ax-1ne0 10811 ax-1rid 10812 ax-rnegex 10813 ax-rrecex 10814 ax-cnre 10815 ax-pre-lttri 10816 ax-pre-lttrn 10817 ax-pre-ltadd 10818 ax-pre-mulgt0 10819 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-pss 3894 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-tp 4555 df-op 4557 df-uni 4829 df-iun 4915 df-br 5063 df-opab 5125 df-mpt 5145 df-tr 5171 df-id 5464 df-eprel 5469 df-po 5477 df-so 5478 df-fr 5518 df-we 5520 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-pred 6169 df-ord 6225 df-on 6226 df-lim 6227 df-suc 6228 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-riota 7179 df-ov 7225 df-oprab 7226 df-mpo 7227 df-om 7654 df-wrecs 8056 df-recs 8117 df-rdg 8155 df-er 8400 df-en 8636 df-dom 8637 df-sdom 8638 df-pnf 10882 df-mnf 10883 df-xr 10884 df-ltxr 10885 df-le 10886 df-sub 11077 df-neg 11078 df-nn 11844 df-2 11906 df-3 11907 df-4 11908 df-5 11909 df-6 11910 df-7 11911 df-8 11912 df-9 11913 df-dec 12307 df-sets 16730 df-slot 16748 df-ndx 16758 df-base 16774 df-ple 16835 df-odu 17808 df-proset 17815 df-poset 17833 |
This theorem is referenced by: odulatb 17953 oduclatb 18026 |
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