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Mirrors > Home > MPE Home > Th. List > df-odu | Structured version Visualization version GIF version |
Description: Define the dual of an
ordered structure, which replaces the order
component of the structure with its reverse. See odubas 18291, oduleval 18289,
and oduleg 18290 for its principal properties.
EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18495. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
Ref | Expression |
---|---|
df-odu | ⊢ ODual = (𝑤 ∈ V ↦ (𝑤 sSet 〈(le‘ndx), ◡(le‘𝑤)〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | codu 18286 | . 2 class ODual | |
2 | vw | . . 3 setvar 𝑤 | |
3 | cvv 3461 | . . 3 class V | |
4 | 2 | cv 1532 | . . . 4 class 𝑤 |
5 | cnx 17170 | . . . . . 6 class ndx | |
6 | cple 17248 | . . . . . 6 class le | |
7 | 5, 6 | cfv 6549 | . . . . 5 class (le‘ndx) |
8 | 4, 6 | cfv 6549 | . . . . . 6 class (le‘𝑤) |
9 | 8 | ccnv 5677 | . . . . 5 class ◡(le‘𝑤) |
10 | 7, 9 | cop 4636 | . . . 4 class 〈(le‘ndx), ◡(le‘𝑤)〉 |
11 | csts 17140 | . . . 4 class sSet | |
12 | 4, 10, 11 | co 7419 | . . 3 class (𝑤 sSet 〈(le‘ndx), ◡(le‘𝑤)〉) |
13 | 2, 3, 12 | cmpt 5232 | . 2 class (𝑤 ∈ V ↦ (𝑤 sSet 〈(le‘ndx), ◡(le‘𝑤)〉)) |
14 | 1, 13 | wceq 1533 | 1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet 〈(le‘ndx), ◡(le‘𝑤)〉)) |
Colors of variables: wff setvar class |
This definition is referenced by: oduval 18288 |
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