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Theorem bj-elccinfty 32734
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 32727 . . . . 5 inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
21funmpt2 5885 . . . 4 Fun inftyexpi
32jctl 563 . . 3 (𝐴 ∈ dom inftyexpi → (Fun inftyexpi ∧ 𝐴 ∈ dom inftyexpi ))
4 opex 4893 . . . . 5 𝑥, ℂ⟩ ∈ V
54, 1dmmpti 5980 . . . 4 dom inftyexpi = (-π(,]π)
65eqcomi 2630 . . 3 (-π(,]π) = dom inftyexpi
73, 6eleq2s 2716 . 2 (𝐴 ∈ (-π(,]π) → (Fun inftyexpi ∧ 𝐴 ∈ dom inftyexpi ))
8 fvelrn 6308 . 2 ((Fun inftyexpi ∧ 𝐴 ∈ dom inftyexpi ) → (inftyexpi ‘𝐴) ∈ ran inftyexpi )
9 df-bj-ccinfty 32732 . . . . 5 = ran inftyexpi
109eqcomi 2630 . . . 4 ran inftyexpi = ℂ
1110eleq2i 2690 . . 3 ((inftyexpi ‘𝐴) ∈ ran inftyexpi ↔ (inftyexpi ‘𝐴) ∈ ℂ)
1211biimpi 206 . 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 384  wcel 1987  cop 4154  dom cdm 5074  ran crn 5075  Fun wfun 5841  cfv 5847  (class class class)co 6604  cc 9878  -cneg 10211  (,]cioc 12118  πcpi 14722  inftyexpi cinftyexpi 32726  cccinfty 32731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-bj-inftyexpi 32727  df-bj-ccinfty 32732
This theorem is referenced by:  bj-pinftyccb  32741
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