Theorem bj-elccinfty 34498

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 34491	. . . . 5 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
2	1	funmpt2 6396	. . . 4 ⊢ Fun +∞ei
3	2	jctl 526	. . 3 ⊢ (𝐴 ∈ dom +∞ei → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei))
4		opex 5358	. . . . 5 ⊢ ⟨𝑥, ℂ⟩ ∈ V
5	4, 1	dmmpti 6494	. . . 4 ⊢ dom +∞ei = (-π(,]π)
6	5	eqcomi 2832	. . 3 ⊢ (-π(,]π) = dom +∞ei
7	3, 6	eleq2s 2933	. 2 ⊢ (𝐴 ∈ (-π(,]π) → (Fun +∞ei ∧ 𝐴 ∈ dom +∞ei))
8		fvelrn 6846	. 2 ⊢ ((Fun +∞ei ∧ 𝐴 ∈ dom +∞ei) → (+∞ei‘𝐴) ∈ ran +∞ei)
9		df-bj-ccinfty 34496	. . . . 5 ⊢ ℂ∞ = ran +∞ei
10	9	eqcomi 2832	. . . 4 ⊢ ran +∞ei = ℂ∞
11	10	eleq2i 2906	. . 3 ⊢ ((+∞ei‘𝐴) ∈ ran +∞ei ↔ (+∞ei‘𝐴) ∈ ℂ∞)
12	11	biimpi 218	. 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 398   ∈ wcel 2114  ⟨cop 4575  dom cdm 5557  ran crn 5558  Fun wfun 6351  ‘cfv 6357  (class class class)co 7158  ℂcc 10537  -cneg 10873  (,]cioc 12742  πcpi 15422  +∞eⁱcinftyexpi 34490  ℂ_∞cccinfty 34495

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-fv 6365  df-bj-inftyexpi 34491  df-bj-ccinfty 34496

This theorem is referenced by:  bj-pinftyccb  34505