Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 17739 | . 2 ⊢ (𝑂 ∈ Poset → 𝐷 ∈ Poset) |
3 | eqid 2821 | . . . 4 ⊢ (ODual‘𝐷) = (ODual‘𝐷) | |
4 | 3 | odupos 17739 | . . 3 ⊢ (𝐷 ∈ Poset → (ODual‘𝐷) ∈ Poset) |
5 | fvexd 6679 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (ODual‘𝐷) ∈ V) | |
6 | id 22 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝑉) | |
7 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
8 | 1, 7 | odubas 17737 | . . . . . 6 ⊢ (Base‘𝑂) = (Base‘𝐷) |
9 | 3, 8 | odubas 17737 | . . . . 5 ⊢ (Base‘𝑂) = (Base‘(ODual‘𝐷)) |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (Base‘𝑂) = (Base‘(ODual‘𝐷))) |
11 | eqidd 2822 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (Base‘𝑂) = (Base‘𝑂)) | |
12 | eqid 2821 | . . . . . . . . . 10 ⊢ (le‘𝑂) = (le‘𝑂) | |
13 | 1, 12 | oduleval 17735 | . . . . . . . . 9 ⊢ ◡(le‘𝑂) = (le‘𝐷) |
14 | 3, 13 | oduleval 17735 | . . . . . . . 8 ⊢ ◡◡(le‘𝑂) = (le‘(ODual‘𝐷)) |
15 | 14 | eqcomi 2830 | . . . . . . 7 ⊢ (le‘(ODual‘𝐷)) = ◡◡(le‘𝑂) |
16 | 15 | breqi 5064 | . . . . . 6 ⊢ (𝑎(le‘(ODual‘𝐷))𝑏 ↔ 𝑎◡◡(le‘𝑂)𝑏) |
17 | vex 3497 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
18 | vex 3497 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
19 | 17, 18 | brcnv 5747 | . . . . . 6 ⊢ (𝑎◡◡(le‘𝑂)𝑏 ↔ 𝑏◡(le‘𝑂)𝑎) |
20 | 18, 17 | brcnv 5747 | . . . . . 6 ⊢ (𝑏◡(le‘𝑂)𝑎 ↔ 𝑎(le‘𝑂)𝑏) |
21 | 16, 19, 20 | 3bitri 299 | . . . . 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 17738 | . . 3 ⊢ (𝑂 ∈ 𝑉 → ((ODual‘𝐷) ∈ Poset ↔ 𝑂 ∈ Poset)) |
24 | 4, 23 | syl5ib 246 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝐷 ∈ Poset → 𝑂 ∈ Poset)) |
25 | 2, 24 | impbid2 228 | 1 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 class class class wbr 5058 ◡ccnv 5548 ‘cfv 6349 Basecbs 16477 lecple 16566 Posetcpo 17544 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 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 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-odu 17733 |
This theorem is referenced by: odulatb 17747 oduclatb 17748 |
Copyright terms: Public domain | W3C validator |