Theorem bj-elccinfty 37746

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

Assertion

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

Proof of Theorem bj-elccinfty

Step	Hyp	Ref	Expression
1		df-bj-inftyexpi 37739	. . . . 5 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
2	1	funmpt2 6576	. . . 4 ⊢ Fun +∞ei
3	2	jctl 532	. . 3 ⊢ (𝐴 ∈ dom +∞ei → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei))
4		opex 5446	. . . . 5 ⊢ ⟨𝑥, ℂ⟩ ∈ V
5	4, 1	dmmpti 6680	. . . 4 ⊢ dom +∞ei = (-π(,]π)
6	5	eqcomi 2778	. . 3 ⊢ (-π(,]π) = dom +∞ei
7	3, 6	eleq2s 2887	. 2 ⊢ (𝐴 ∈ (-π(,]π) → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei))
8		fvelrn 7072	. 2 ⊢ ((Fun +∞ei ∧ 𝐴 ∈ dom +∞ei) → (+∞ei‘𝐴) ∈ ran +∞ei)
9		df-bj-ccinfty 37744	. . . . 5 ⊢ ℂ∞ = ran +∞ei
10	9	eqcomi 2778	. . . 4 ⊢ ran +∞ei = ℂ∞
11	10	eleq2i 2861	. . 3 ⊢ ((+∞ei‘𝐴) ∈ ran +∞ei ↔ (+∞ei‘𝐴) ∈ ℂ∞)
12	11	biimpi 219	. 2 ⊢ ((+∞ei‘𝐴) ∈ ran +∞ei → (+∞ei‘𝐴) ∈ ℂ∞)
13	7, 8, 12	3syl 19	1 ⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) ∈ ℂ∞)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2149  ⟨cop 4600  dom cdm 5662  ran crn 5663  Fun wfun 6531  ‘cfv 6537  (class class class)co 7411  ℂcc 11098  -cneg 11442  (,]cioc 13373  πcpi 16120  +∞eⁱcinftyexpi 37738  ℂ_∞cccinfty 37743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-bj-inftyexpi 37739  df-bj-ccinfty 37744

This theorem is referenced by:  bj-pinftyccb  37753