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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 12880 for its value, uzssz 12899 for its relationship to ℤ, nnuz 12921 and nn0uz 12920 for its relationships to ℕ and ℕ0, and eluz1 12882 and eluz2 12884 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 12878 | . 2 class ℤ≥ | |
| 2 | vj | . . 3 setvar 𝑗 | |
| 3 | cz 12613 | . . 3 class ℤ | |
| 4 | 2 | cv 1539 | . . . . 5 class 𝑗 |
| 5 | vk | . . . . . 6 setvar 𝑘 | |
| 6 | 5 | cv 1539 | . . . . 5 class 𝑘 |
| 7 | cle 11296 | . . . . 5 class ≤ | |
| 8 | 4, 6, 7 | wbr 5143 | . . . 4 wff 𝑗 ≤ 𝑘 |
| 9 | 8, 5, 3 | crab 3436 | . . 3 class {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} |
| 10 | 2, 3, 9 | cmpt 5225 | . 2 class (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
| 11 | 1, 10 | wceq 1540 | 1 wff ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
| Colors of variables: wff setvar class |
| This definition is referenced by: uzval 12880 uzf 12881 |
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