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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 11689 for its value, uzssz 11707 for its relationship to ℤ, nnuz 11723 and nn0uz 11722 for its relationships to ℕ and ℕ0, and eluz1 11691 and eluz2 11693 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 11687 | . 2 class ℤ≥ | |
2 | vj | . . 3 setvar 𝑗 | |
3 | cz 11377 | . . 3 class ℤ | |
4 | 2 | cv 1482 | . . . . 5 class 𝑗 |
5 | vk | . . . . . 6 setvar 𝑘 | |
6 | 5 | cv 1482 | . . . . 5 class 𝑘 |
7 | cle 10075 | . . . . 5 class ≤ | |
8 | 4, 6, 7 | wbr 4653 | . . . 4 wff 𝑗 ≤ 𝑘 |
9 | 8, 5, 3 | crab 2916 | . . 3 class {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} |
10 | 2, 3, 9 | cmpt 4729 | . 2 class (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
11 | 1, 10 | wceq 1483 | 1 wff ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
Colors of variables: wff setvar class |
This definition is referenced by: uzval 11689 uzf 11690 |
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