Theorem bj-elccinfty 33101

Description: A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)

Assertion

Ref	Expression
bj-elccinfty	⊢ (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) ∈ ℂ∞)

Proof of Theorem bj-elccinfty

Step	Hyp	Ref	Expression
1		df-bj-inftyexpi 33094	. . . . 5 ⊢ inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
2	1	funmpt2 5927	. . . 4 ⊢ Fun inftyexpi
3	2	jctl 564	. . 3 ⊢ (𝐴 ∈ dom inftyexpi → (Fun inftyexpi ∧ 𝐴 ∈ dom inftyexpi ))
4		opex 4932	. . . . 5 ⊢ ⟨𝑥, ℂ⟩ ∈ V
5	4, 1	dmmpti 6023	. . . 4 ⊢ dom inftyexpi = (-π(,]π)
6	5	eqcomi 2631	. . 3 ⊢ (-π(,]π) = dom inftyexpi
7	3, 6	eleq2s 2719	. 2 ⊢ (𝐴 ∈ (-π(,]π) → (Fun inftyexpi ∧ 𝐴 ∈ dom inftyexpi ))
8		fvelrn 6352	. 2 ⊢ ((Fun inftyexpi ∧ 𝐴 ∈ dom inftyexpi ) → (inftyexpi ‘𝐴) ∈ ran inftyexpi )
9		df-bj-ccinfty 33099	. . . . 5 ⊢ ℂ∞ = ran inftyexpi
10	9	eqcomi 2631	. . . 4 ⊢ ran inftyexpi = ℂ∞
11	10	eleq2i 2693	. . 3 ⊢ ((inftyexpi ‘𝐴) ∈ ran inftyexpi ↔ (inftyexpi ‘𝐴) ∈ ℂ∞)
12	11	biimpi 206	. 2 ⊢ ((inftyexpi ‘𝐴) ∈ ran inftyexpi → (inftyexpi ‘𝐴) ∈ ℂ∞)
13	7, 8, 12	3syl 18	1 ⊢ (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) ∈ ℂ∞)

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990  ⟨cop 4183  dom cdm 5114  ran crn 5115  Fun wfun 5882  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  -cneg 10267  (,]cioc 12176  πcpi 14797  inftyexpi cinftyexpi 33093  ℂ_∞cccinfty 33098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-bj-inftyexpi 33094  df-bj-ccinfty 33099

This theorem is referenced by:  bj-pinftyccb  33108