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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 12854 for its value, uzssz 12873 for its relationship to ℤ, nnuz 12895 and nn0uz 12894 for its relationships to ℕ and ℕ0, and eluz1 12856 and eluz2 12858 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 12852 | . 2 class ℤ≥ | |
2 | vj | . . 3 setvar 𝑗 | |
3 | cz 12588 | . . 3 class ℤ | |
4 | 2 | cv 1533 | . . . . 5 class 𝑗 |
5 | vk | . . . . . 6 setvar 𝑘 | |
6 | 5 | cv 1533 | . . . . 5 class 𝑘 |
7 | cle 11279 | . . . . 5 class ≤ | |
8 | 4, 6, 7 | wbr 5148 | . . . 4 wff 𝑗 ≤ 𝑘 |
9 | 8, 5, 3 | crab 3429 | . . 3 class {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} |
10 | 2, 3, 9 | cmpt 5231 | . 2 class (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
11 | 1, 10 | wceq 1534 | 1 wff ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
Colors of variables: wff setvar class |
This definition is referenced by: uzval 12854 uzf 12855 |
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