Theorem bj-elccinfty 38135

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 38128	. . . . 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 5432	. . . . 5 ⊢ ⟨𝑥, ℂ⟩ ∈ V
5	4, 1	dmmpti 6683	. . . 4 ⊢ dom +∞ei = ( -π(,]π)
6	5	eqcomi 2770	. . 3 ⊢ ( -π(,]π) = dom +∞ei
7	3, 6	eleq2s 2879	. 2 ⊢ (𝐴 ∈ ( -π(,]π) → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei))
8		fvelrn 7076	. 2 ⊢ ((Fun +∞ei ∧ 𝐴 ∈ dom +∞ei) → (+∞ei‘𝐴) ∈ ran +∞ei)
9		df-bj-ccinfty 38133	. . . . 5 ⊢ ℂ∞ = ran +∞ei
10	9	eqcomi 2770	. . . 4 ⊢ ran +∞ei = ℂ∞
11	10	eleq2i 2853	. . 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 5651  ran crn 5652  Fun wfun 6532  ‘cfv 6538  (class class class)co 7420  ℂcc 11198   -cneg 11542  (,]cioc 13477  πcpi 16232  +∞eⁱcinftyexpi 38127  ℂ_∞cccinfty 38132

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 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-bj-inftyexpi 38128  df-bj-ccinfty 38133

This theorem is used by:  bj-pinftyccb  38142