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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
StepHypRef Expression
1 df-bj-inftyexpi 32271 . . . . 5 inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
21funmpt2 5841 . . . 4 Fun inftyexpi
32jctl 562 . . 3 (𝐴 ∈ dom inftyexpi → (Fun inftyexpi ∧ 𝐴 ∈ dom inftyexpi ))
4 opex 4859 . . . . 5 𝑥, ℂ⟩ ∈ V
54, 1dmmpti 5936 . . . 4 dom inftyexpi = (-π(,]π)
65eqcomi 2619 . . 3 (-π(,]π) = dom inftyexpi
73, 6eleq2s 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 = ran inftyexpi
109eqcomi 2619 . . . 4 ran inftyexpi = ℂ
1110eleq2i 2680 . . 3 ((inftyexpi ‘𝐴) ∈ ran inftyexpi ↔ (inftyexpi ‘𝐴) ∈ ℂ)
1211biimpi 205 . 2 ((inftyexpi ‘𝐴) ∈ ran inftyexpi → (inftyexpi ‘𝐴) ∈ ℂ)
137, 8, 123syl 18 1 (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) ∈ ℂ)
 Colors of variables: wff setvar class 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
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