Theorem bj-elccinfty 32278

Description: A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)

Assertion

Ref	Expression
bj-elccinfty	⊢ (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) ∈ ℂ∞)

Proof of Theorem bj-elccinfty

Step	Hyp	Ref	Expression
1		df-bj-inftyexpi 32271	. . . . 5 ⊢ inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
2	1	funmpt2 5841	. . . 4 ⊢ Fun inftyexpi
3	2	jctl 562	. . 3 ⊢ (𝐴 ∈ dom inftyexpi → (Fun inftyexpi ∧ 𝐴 ∈ dom inftyexpi ))
4		opex 4859	. . . . 5 ⊢ ⟨𝑥, ℂ⟩ ∈ V
5	4, 1	dmmpti 5936	. . . 4 ⊢ dom inftyexpi = (-π(,]π)
6	5	eqcomi 2619	. . 3 ⊢ (-π(,]π) = dom inftyexpi
7	3, 6	eleq2s 2706	. 2 ⊢ (𝐴 ∈ (-π(,]π) → (Fun inftyexpi ∧ 𝐴 ∈ dom inftyexpi ))
8		fvelrn 6260	. 2 ⊢ ((Fun inftyexpi ∧ 𝐴 ∈ dom inftyexpi ) → (inftyexpi ‘𝐴) ∈ ran inftyexpi )
9		df-bj-ccinfty 32276	. . . . 5 ⊢ ℂ∞ = ran inftyexpi
10	9	eqcomi 2619	. . . 4 ⊢ ran inftyexpi = ℂ∞
11	10	eleq2i 2680	. . 3 ⊢ ((inftyexpi ‘𝐴) ∈ ran inftyexpi ↔ (inftyexpi ‘𝐴) ∈ ℂ∞)
12	11	biimpi 205	. 2 ⊢ ((inftyexpi ‘𝐴) ∈ ran inftyexpi → (inftyexpi ‘𝐴) ∈ ℂ∞)
13	7, 8, 12	3syl 18	1 ⊢ (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) ∈ ℂ∞)

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ⟨cop 4131  dom cdm 5038  ran crn 5039  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  -cneg 10146  (,]cioc 12047  πcpi 14636  inftyexpi cinftyexpi 32270  ℂ_∞cccinfty 32275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-bj-inftyexpi 32271  df-bj-ccinfty 32276

This theorem is referenced by:  bj-pinftyccb  32285