Definition df-bj-inftyexpi 37201

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.)

Assertion

Ref	Expression
df-bj-inftyexpi	⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)

Detailed syntax breakdown of Definition df-bj-inftyexpi

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⟩)

This definition is referenced by:  bj-inftyexpiinv  37202  bj-inftyexpidisj  37204  bj-ccinftydisj  37207  bj-elccinfty  37208  bj-minftyccb  37219