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Theorem bj-elccinfty 37157
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 37150 . . . . 5 +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
21funmpt2 6602 . . . 4 Fun +∞ei
32jctl 523 . . 3 (𝐴 ∈ dom +∞ei → (Fun +∞ei𝐴 ∈ dom +∞ei))
4 opex 5467 . . . . 5 𝑥, ℂ⟩ ∈ V
54, 1dmmpti 6708 . . . 4 dom +∞ei = (-π(,]π)
65eqcomi 2742 . . 3 (-π(,]π) = dom +∞ei
73, 6eleq2s 2855 . 2 (𝐴 ∈ (-π(,]π) → (Fun +∞ei𝐴 ∈ dom +∞ei))
8 fvelrn 7090 . 2 ((Fun +∞ei𝐴 ∈ dom +∞ei) → (+∞ei𝐴) ∈ ran +∞ei)
9 df-bj-ccinfty 37155 . . . . 5 = ran +∞ei
109eqcomi 2742 . . . 4 ran +∞ei = ℂ
1110eleq2i 2829 . . 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 2104  cop 4636  dom cdm 5683  ran crn 5684  Fun wfun 6552  cfv 6558  (class class class)co 7425  cc 11144  -cneg 11484  (,]cioc 13378  πcpi 16088  +∞eicinftyexpi 37149  cccinfty 37154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-iota 6510  df-fun 6560  df-fn 6561  df-fv 6566  df-bj-inftyexpi 37150  df-bj-ccinfty 37155
This theorem is referenced by:  bj-pinftyccb  37164
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