Theorem bj-elccinfty 37358

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 37351	. . . . 5 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
2	1	funmpt2 6529	. . . 4 ⊢ Fun +∞ei
3	2	jctl 523	. . 3 ⊢ (𝐴 ∈ dom +∞ei → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei))
4		opex 5410	. . . . 5 ⊢ ⟨𝑥, ℂ⟩ ∈ V
5	4, 1	dmmpti 6634	. . . 4 ⊢ dom +∞ei = (-π(,]π)
6	5	eqcomi 2743	. . 3 ⊢ (-π(,]π) = dom +∞ei
7	3, 6	eleq2s 2852	. 2 ⊢ (𝐴 ∈ (-π(,]π) → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei))
8		fvelrn 7019	. 2 ⊢ ((Fun +∞ei ∧ 𝐴 ∈ dom +∞ei) → (+∞ei‘𝐴) ∈ ran +∞ei)
9		df-bj-ccinfty 37356	. . . . 5 ⊢ ℂ∞ = ran +∞ei
10	9	eqcomi 2743	. . . 4 ⊢ ran +∞ei = ℂ∞
11	10	eleq2i 2826	. . 3 ⊢ ((+∞ei‘𝐴) ∈ ran +∞ei ↔ (+∞ei‘𝐴) ∈ ℂ∞)
12	11	biimpi 216	. 2 ⊢ ((+∞ei‘𝐴) ∈ ran +∞ei → (+∞ei‘𝐴) ∈ ℂ∞)
13	7, 8, 12	3syl 18	1 ⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) ∈ ℂ∞)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  ⟨cop 4584  dom cdm 5622  ran crn 5623  Fun wfun 6484  ‘cfv 6490  (class class class)co 7356  ℂcc 11022  -cneg 11363  (,]cioc 13260  πcpi 15987  +∞eⁱcinftyexpi 37350  ℂ_∞cccinfty 37355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-fv 6498  df-bj-inftyexpi 37351  df-bj-ccinfty 37356

This theorem is referenced by:  bj-pinftyccb  37365