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Theorem bj-elccinfty 37193
Description: A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
Assertion
Ref Expression
bj-elccinfty (𝐴 ∈ (-π(,]π) → (+∞ei𝐴) ∈ ℂ)

Proof of Theorem bj-elccinfty
StepHypRef Expression
1 df-bj-inftyexpi 37186 . . . . 5 +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
21funmpt2 6603 . . . 4 Fun +∞ei
32jctl 523 . . 3 (𝐴 ∈ dom +∞ei → (Fun +∞ei𝐴 ∈ dom +∞ei))
4 opex 5467 . . . . 5 𝑥, ℂ⟩ ∈ V
54, 1dmmpti 6710 . . . 4 dom +∞ei = (-π(,]π)
65eqcomi 2745 . . 3 (-π(,]π) = dom +∞ei
73, 6eleq2s 2858 . 2 (𝐴 ∈ (-π(,]π) → (Fun +∞ei𝐴 ∈ dom +∞ei))
8 fvelrn 7094 . 2 ((Fun +∞ei𝐴 ∈ dom +∞ei) → (+∞ei𝐴) ∈ ran +∞ei)
9 df-bj-ccinfty 37191 . . . . 5 = ran +∞ei
109eqcomi 2745 . . . 4 ran +∞ei = ℂ
1110eleq2i 2832 . . 3 ((+∞ei𝐴) ∈ ran +∞ei ↔ (+∞ei𝐴) ∈ ℂ)
1211biimpi 216 . 2 ((+∞ei𝐴) ∈ ran +∞ei → (+∞ei𝐴) ∈ ℂ)
137, 8, 123syl 18 1 (𝐴 ∈ (-π(,]π) → (+∞ei𝐴) ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  cop 4630  dom cdm 5683  ran crn 5684  Fun wfun 6553  cfv 6559  (class class class)co 7429  cc 11149  -cneg 11489  (,]cioc 13384  πcpi 16098  +∞eicinftyexpi 37185  cccinfty 37190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-iota 6512  df-fun 6561  df-fn 6562  df-fv 6567  df-bj-inftyexpi 37186  df-bj-ccinfty 37191
This theorem is referenced by:  bj-pinftyccb  37200
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