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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 17135 | . 2 ⊢ (𝑂 ∈ Poset → 𝐷 ∈ Poset) |
3 | eqid 2622 | . . . 4 ⊢ (ODual‘𝐷) = (ODual‘𝐷) | |
4 | 3 | odupos 17135 | . . 3 ⊢ (𝐷 ∈ Poset → (ODual‘𝐷) ∈ Poset) |
5 | fvexd 6203 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (ODual‘𝐷) ∈ V) | |
6 | id 22 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝑉) | |
7 | eqid 2622 | . . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
8 | 1, 7 | odubas 17133 | . . . . . 6 ⊢ (Base‘𝑂) = (Base‘𝐷) |
9 | 3, 8 | odubas 17133 | . . . . 5 ⊢ (Base‘𝑂) = (Base‘(ODual‘𝐷)) |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (Base‘𝑂) = (Base‘(ODual‘𝐷))) |
11 | eqidd 2623 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (Base‘𝑂) = (Base‘𝑂)) | |
12 | eqid 2622 | . . . . . . . . . 10 ⊢ (le‘𝑂) = (le‘𝑂) | |
13 | 1, 12 | oduleval 17131 | . . . . . . . . 9 ⊢ ◡(le‘𝑂) = (le‘𝐷) |
14 | 3, 13 | oduleval 17131 | . . . . . . . 8 ⊢ ◡◡(le‘𝑂) = (le‘(ODual‘𝐷)) |
15 | 14 | eqcomi 2631 | . . . . . . 7 ⊢ (le‘(ODual‘𝐷)) = ◡◡(le‘𝑂) |
16 | 15 | breqi 4659 | . . . . . 6 ⊢ (𝑎(le‘(ODual‘𝐷))𝑏 ↔ 𝑎◡◡(le‘𝑂)𝑏) |
17 | vex 3203 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
18 | vex 3203 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
19 | 17, 18 | brcnv 5305 | . . . . . 6 ⊢ (𝑎◡◡(le‘𝑂)𝑏 ↔ 𝑏◡(le‘𝑂)𝑎) |
20 | 18, 17 | brcnv 5305 | . . . . . 6 ⊢ (𝑏◡(le‘𝑂)𝑎 ↔ 𝑎(le‘𝑂)𝑏) |
21 | 16, 19, 20 | 3bitri 286 | . . . . 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 17134 | . . 3 ⊢ (𝑂 ∈ 𝑉 → ((ODual‘𝐷) ∈ Poset ↔ 𝑂 ∈ Poset)) |
24 | 4, 23 | syl5ib 234 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝐷 ∈ Poset → 𝑂 ∈ Poset)) |
25 | 2, 24 | impbid2 216 | 1 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 ◡ccnv 5113 ‘cfv 5888 Basecbs 15857 lecple 15948 Posetcpo 16940 ODualcodu 17128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-dec 11494 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ple 15961 df-preset 16928 df-poset 16946 df-odu 17129 |
This theorem is referenced by: odulatb 17143 oduclatb 17144 |
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