Theorem bj-elccinfty 37969

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 37962	. . . . 5 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
2	1	funmpt2 6573	. . . 4 ⊢ Fun +∞ei
3	2	jctl 533	. . 3 ⊢ (𝐴 ∈ dom +∞ei → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei))
4		opex 5439	. . . . 5 ⊢ ⟨𝑥, ℂ⟩ ∈ V
5	4, 1	dmmpti 6677	. . . 4 ⊢ dom +∞ei = (-π(,]π)
6	5	eqcomi 2769	. . 3 ⊢ (-π(,]π) = dom +∞ei
7	3, 6	eleq2s 2878	. 2 ⊢ (𝐴 ∈ (-π(,]π) → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei))
8		fvelrn 7070	. 2 ⊢ ((Fun +∞ei ∧ 𝐴 ∈ dom +∞ei) → (+∞ei‘𝐴) ∈ ran +∞ei)
9		df-bj-ccinfty 37967	. . . . 5 ⊢ ℂ∞ = ran +∞ei
10	9	eqcomi 2769	. . . 4 ⊢ ran +∞ei = ℂ∞
11	10	eleq2i 2852	. . 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 2145  ⟨cop 4590  dom cdm 5655  ran crn 5656  Fun wfun 6527  ‘cfv 6533  (class class class)co 7414  ℂcc 11125  -cneg 11469  (,]cioc 13402  πcpi 16155  +∞eⁱcinftyexpi 37961  ℂ_∞cccinfty 37966

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541  df-bj-inftyexpi 37962  df-bj-ccinfty 37967

This theorem is used by:  bj-pinftyccb  37976