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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 11565 for its value, uzssz 11583 for its relationship to ℤ, nnuz 11599 and nn0uz 11598 for its relationships to ℕ and ℕ0, and eluz1 11567 and eluz2 11569 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 11563 | . 2 class ℤ≥ | |
2 | vj | . . 3 setvar 𝑗 | |
3 | cz 11254 | . . 3 class ℤ | |
4 | 2 | cv 1474 | . . . . 5 class 𝑗 |
5 | vk | . . . . . 6 setvar 𝑘 | |
6 | 5 | cv 1474 | . . . . 5 class 𝑘 |
7 | cle 9954 | . . . . 5 class ≤ | |
8 | 4, 6, 7 | wbr 4583 | . . . 4 wff 𝑗 ≤ 𝑘 |
9 | 8, 5, 3 | crab 2900 | . . 3 class {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} |
10 | 2, 3, 9 | cmpt 4643 | . 2 class (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
11 | 1, 10 | wceq 1475 | 1 wff ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
Colors of variables: wff setvar class |
This definition is referenced by: uzval 11565 uzf 11566 |
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