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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-lhyp | Structured version Visualization version GIF version |
Description: Define the set of lattice hyperplanes, which are all lattice elements covered by 1 (i.e., all co-atoms). We call them "hyperplanes" instead of "co-atoms" in analogy with projective geometry hyperplanes. (Contributed by NM, 11-May-2012.) |
Ref | Expression |
---|---|
df-lhyp | ⊢ LHyp = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ 𝑥( ⋖ ‘𝑘)(1.‘𝑘)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clh 37740 | . 2 class LHyp | |
2 | vk | . . 3 setvar 𝑘 | |
3 | cvv 3413 | . . 3 class V | |
4 | vx | . . . . . 6 setvar 𝑥 | |
5 | 4 | cv 1542 | . . . . 5 class 𝑥 |
6 | 2 | cv 1542 | . . . . . 6 class 𝑘 |
7 | cp1 17935 | . . . . . 6 class 1. | |
8 | 6, 7 | cfv 6385 | . . . . 5 class (1.‘𝑘) |
9 | ccvr 37018 | . . . . . 6 class ⋖ | |
10 | 6, 9 | cfv 6385 | . . . . 5 class ( ⋖ ‘𝑘) |
11 | 5, 8, 10 | wbr 5058 | . . . 4 wff 𝑥( ⋖ ‘𝑘)(1.‘𝑘) |
12 | cbs 16765 | . . . . 5 class Base | |
13 | 6, 12 | cfv 6385 | . . . 4 class (Base‘𝑘) |
14 | 11, 4, 13 | crab 3065 | . . 3 class {𝑥 ∈ (Base‘𝑘) ∣ 𝑥( ⋖ ‘𝑘)(1.‘𝑘)} |
15 | 2, 3, 14 | cmpt 5140 | . 2 class (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ 𝑥( ⋖ ‘𝑘)(1.‘𝑘)}) |
16 | 1, 15 | wceq 1543 | 1 wff LHyp = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ 𝑥( ⋖ ‘𝑘)(1.‘𝑘)}) |
Colors of variables: wff setvar class |
This definition is referenced by: lhpset 37751 |
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