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Mirrors > Home > MPE Home > Th. List > df-uz | Structured version Visualization version GIF version |
Description: Define a function whose value at 𝑗 is the semi-infinite set of contiguous integers starting at 𝑗, which we will also call the upper integers starting at 𝑗. Read "ℤ≥‘𝑀 " as "the set of integers greater than or equal to 𝑀". See uzval 12246 for its value, uzssz 12265 for its relationship to ℤ, nnuz 12282 and nn0uz 12281 for its relationships to ℕ and ℕ0, and eluz1 12248 and eluz2 12250 for its membership relations. (Contributed by NM, 5-Sep-2005.) |
Ref | Expression |
---|---|
df-uz | ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cuz 12244 | . 2 class ℤ≥ | |
2 | vj | . . 3 setvar 𝑗 | |
3 | cz 11982 | . . 3 class ℤ | |
4 | 2 | cv 1536 | . . . . 5 class 𝑗 |
5 | vk | . . . . . 6 setvar 𝑘 | |
6 | 5 | cv 1536 | . . . . 5 class 𝑘 |
7 | cle 10676 | . . . . 5 class ≤ | |
8 | 4, 6, 7 | wbr 5066 | . . . 4 wff 𝑗 ≤ 𝑘 |
9 | 8, 5, 3 | crab 3142 | . . 3 class {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} |
10 | 2, 3, 9 | cmpt 5146 | . 2 class (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
11 | 1, 10 | wceq 1537 | 1 wff ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
Colors of variables: wff setvar class |
This definition is referenced by: uzval 12246 uzf 12247 |
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