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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 36089. 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 36075 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 36082 | . 2 class +∞ei | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cpi 16009 | . . . . 5 class π | |
4 | 3 | cneg 11444 | . . . 4 class -π |
5 | cioc 13324 | . . . 4 class (,] | |
6 | 4, 3, 5 | co 7408 | . . 3 class (-π(,]π) |
7 | 2 | cv 1540 | . . . 4 class 𝑥 |
8 | cc 11107 | . . . 4 class ℂ | |
9 | 7, 8 | cop 4634 | . . 3 class ⟨𝑥, ℂ⟩ |
10 | 2, 6, 9 | cmpt 5231 | . 2 class (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩) |
11 | 1, 10 | wceq 1541 | 1 wff +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩) |
Colors of variables: wff setvar class |
This definition is referenced by: bj-inftyexpiinv 36084 bj-inftyexpidisj 36086 bj-ccinftydisj 36089 bj-elccinfty 36090 bj-minftyccb 36101 |
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