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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 11881 for its value, uzssz 11899 for its relationship to ℤ, nnuz 11916 and nn0uz 11915 for its relationships to ℕ and ℕ0, and eluz1 11883 and eluz2 11885 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 11879 | . 2 class ℤ≥ | |
2 | vj | . . 3 setvar 𝑗 | |
3 | cz 11569 | . . 3 class ℤ | |
4 | 2 | cv 1631 | . . . . 5 class 𝑗 |
5 | vk | . . . . . 6 setvar 𝑘 | |
6 | 5 | cv 1631 | . . . . 5 class 𝑘 |
7 | cle 10267 | . . . . 5 class ≤ | |
8 | 4, 6, 7 | wbr 4804 | . . . 4 wff 𝑗 ≤ 𝑘 |
9 | 8, 5, 3 | crab 3054 | . . 3 class {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} |
10 | 2, 3, 9 | cmpt 4881 | . 2 class (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
11 | 1, 10 | wceq 1632 | 1 wff ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
Colors of variables: wff setvar class |
This definition is referenced by: uzval 11881 uzf 11882 |
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