Theorem bj-inftyexpidisj 33531

Description: An element of the circle at infinity is not a complex number. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.)

Assertion

Ref	Expression
bj-inftyexpidisj	⊢ ¬ (inftyexpi ‘𝐴) ∈ ℂ

Proof of Theorem bj-inftyexpidisj

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4559	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2		df-bj-inftyexpi 33528	. . . . 5 ⊢ inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3		opex 5088	. . . . 5 ⊢ ⟨𝐴, ℂ⟩ ∈ V
4	1, 2, 3	fvmpt 6471	. . . 4 ⊢ (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
5		opex 5088	. . . . 5 ⊢ ⟨𝑥, ℂ⟩ ∈ V
6	5, 2	dmmpti 6201	. . . 4 ⊢ dom inftyexpi = (-π(,]π)
7	4, 6	eleq2s 2862	. . 3 ⊢ (𝐴 ∈ dom inftyexpi → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
8		cnex 10270	. . . . . . 7 ⊢ ℂ ∈ V
9	8	prid2 4453	. . . . . 6 ⊢ ℂ ∈ {𝐴, ℂ}
10		eqid 2765	. . . . . . . 8 ⊢ {𝐴, ℂ} = {𝐴, ℂ}
11	10	olci 892	. . . . . . 7 ⊢ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})
12		elopg 5090	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ℂ ∈ V) → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
13	8, 12	mpan2 682	. . . . . . 7 ⊢ (𝐴 ∈ V → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
14	11, 13	mpbiri 249	. . . . . 6 ⊢ (𝐴 ∈ V → {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩)
15		en3lp 8724	. . . . . . 7 ⊢ ¬ (ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ∧ ⟨𝐴, ℂ⟩ ∈ ℂ)
16	15	bj-imn3ani 33008	. . . . . 6 ⊢ ((ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩) → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
17	9, 14, 16	sylancr 581	. . . . 5 ⊢ (𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
18		opprc1 4583	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, ℂ⟩ = ∅)
19		0ncn 10207	. . . . . . 7 ⊢ ¬ ∅ ∈ ℂ
20		eleq1 2832	. . . . . . 7 ⊢ (⟨𝐴, ℂ⟩ = ∅ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ ∅ ∈ ℂ))
21	19, 20	mtbiri 318	. . . . . 6 ⊢ (⟨𝐴, ℂ⟩ = ∅ → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
22	18, 21	syl 17	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
23	17, 22	pm2.61i 176	. . . 4 ⊢ ¬ ⟨𝐴, ℂ⟩ ∈ ℂ
24		eqcom 2772	. . . . . 6 ⊢ ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ ↔ ⟨𝐴, ℂ⟩ = (inftyexpi ‘𝐴))
25	24	biimpi 207	. . . . 5 ⊢ ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → ⟨𝐴, ℂ⟩ = (inftyexpi ‘𝐴))
26	25	eleq1d 2829	. . . 4 ⊢ ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ (inftyexpi ‘𝐴) ∈ ℂ))
27	23, 26	mtbii 317	. . 3 ⊢ ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
28	7, 27	syl 17	. 2 ⊢ (𝐴 ∈ dom inftyexpi → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
29		ndmfv 6405	. . . 4 ⊢ (¬ 𝐴 ∈ dom inftyexpi → (inftyexpi ‘𝐴) = ∅)
30	29	eleq1d 2829	. . 3 ⊢ (¬ 𝐴 ∈ dom inftyexpi → ((inftyexpi ‘𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
31	19, 30	mtbiri 318	. 2 ⊢ (¬ 𝐴 ∈ dom inftyexpi → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
32	28, 31	pm2.61i 176	1 ⊢ ¬ (inftyexpi ‘𝐴) ∈ ℂ

Syntax hints:  ¬ wn 3   ↔ wb 197   ∨ wo 873   = wceq 1652   ∈ wcel 2155  Vcvv 3350  ∅c0 4079  {csn 4334  {cpr 4336  ⟨cop 4340  dom cdm 5277  ‘cfv 6068  (class class class)co 6842  ℂcc 10187  -cneg 10521  (,]cioc 12378  πcpi 15079  inftyexpi cinftyexpi 33527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-reg 8704  ax-cnex 10245

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fn 6071  df-fv 6076  df-c 10195  df-bj-inftyexpi 33528

This theorem is referenced by:  bj-ccinftydisj  33534  bj-pinftynrr  33543  bj-minftynrr  33547