Theorem bj-elccinfty 37917

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 37910	. . . . 5 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
2	1	funmpt2 6579	. . . 4 ⊢ Fun +∞ei
3	2	jctl 533	. . 3 ⊢ (𝐴 ∈ dom +∞ei → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei))
4		opex 5447	. . . . 5 ⊢ ⟨𝑥, ℂ⟩ ∈ V
5	4, 1	dmmpti 6683	. . . 4 ⊢ dom +∞ei = (-π(,]π)
6	5	eqcomi 2774	. . 3 ⊢ (-π(,]π) = dom +∞ei
7	3, 6	eleq2s 2883	. 2 ⊢ (𝐴 ∈ (-π(,]π) → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei))
8		fvelrn 7075	. 2 ⊢ ((Fun +∞ei ∧ 𝐴 ∈ dom +∞ei) → (+∞ei‘𝐴) ∈ ran +∞ei)
9		df-bj-ccinfty 37915	. . . . 5 ⊢ ℂ∞ = ran +∞ei
10	9	eqcomi 2774	. . . 4 ⊢ ran +∞ei = ℂ∞
11	10	eleq2i 2857	. . 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∞)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  ⟨cop 4597  dom cdm 5663  ran crn 5664  Fun wfun 6534  ‘cfv 6540  (class class class)co 7419  ℂcc 11115  -cneg 11459  (,]cioc 13391  πcpi 16144  +∞eⁱcinftyexpi 37909  ℂ_∞cccinfty 37914

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548  df-bj-inftyexpi 37910  df-bj-ccinfty 37915

This theorem is used by:  bj-pinftyccb  37924