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| Mirrors > Home > MPE Home > Th. List > Mathboxes > df-bj-inftyexpi | Structured version Visualization version GIF version | ||
| Description: Definition of the auxiliary function +∞ei parameterizing the circle at infinity ℂ∞ in ℂ̅. We use coupling with ℂ to simplify the proof of bj-ccinftydisj 37207. It could seem more natural to define +∞ei on all of ℝ, but we want to use only basic functions in the definition of ℂ̅. TODO: transition to df-bj-inftyexpitau 37193 instead. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| df-bj-inftyexpi | ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ 〈𝑥, ℂ〉) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cinftyexpi 37200 | . 2 class +∞ei | |
| 2 | vx | . . 3 setvar 𝑥 | |
| 3 | cpi 16098 | . . . . 5 class π | |
| 4 | 3 | cneg 11489 | . . . 4 class -π |
| 5 | cioc 13384 | . . . 4 class (,] | |
| 6 | 4, 3, 5 | co 7429 | . . 3 class (-π(,]π) |
| 7 | 2 | cv 1539 | . . . 4 class 𝑥 |
| 8 | cc 11149 | . . . 4 class ℂ | |
| 9 | 7, 8 | cop 4630 | . . 3 class 〈𝑥, ℂ〉 |
| 10 | 2, 6, 9 | cmpt 5223 | . 2 class (𝑥 ∈ (-π(,]π) ↦ 〈𝑥, ℂ〉) |
| 11 | 1, 10 | wceq 1540 | 1 wff +∞ei = (𝑥 ∈ (-π(,]π) ↦ 〈𝑥, ℂ〉) |
| Colors of variables: wff setvar class |
| This definition is referenced by: bj-inftyexpiinv 37202 bj-inftyexpidisj 37204 bj-ccinftydisj 37207 bj-elccinfty 37208 bj-minftyccb 37219 |
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