Theorem bj-inftyexpidisj 34625

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	⊢ ¬ (+∞ei‘𝐴) ∈ ℂ

Proof of Theorem bj-inftyexpidisj

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4763	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2		df-bj-inftyexpi 34622	. . . . 5 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3		opex 5321	. . . . 5 ⊢ ⟨𝐴, ℂ⟩ ∈ V
4	1, 2, 3	fvmpt 6745	. . . 4 ⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) = ⟨𝐴, ℂ⟩)
5		opex 5321	. . . . 5 ⊢ ⟨𝑥, ℂ⟩ ∈ V
6	5, 2	dmmpti 6464	. . . 4 ⊢ dom +∞ei = (-π(,]π)
7	4, 6	eleq2s 2908	. . 3 ⊢ (𝐴 ∈ dom +∞ei → (+∞ei‘𝐴) = ⟨𝐴, ℂ⟩)
8		cnex 10607	. . . . . . 7 ⊢ ℂ ∈ V
9	8	prid2 4659	. . . . . 6 ⊢ ℂ ∈ {𝐴, ℂ}
10		eqid 2798	. . . . . . . 8 ⊢ {𝐴, ℂ} = {𝐴, ℂ}
11	10	olci 863	. . . . . . 7 ⊢ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})
12		elopg 5323	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ℂ ∈ V) → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
13	8, 12	mpan2 690	. . . . . . 7 ⊢ (𝐴 ∈ V → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
14	11, 13	mpbiri 261	. . . . . 6 ⊢ (𝐴 ∈ V → {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩)
15		en3lp 9061	. . . . . . 7 ⊢ ¬ (ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ∧ ⟨𝐴, ℂ⟩ ∈ ℂ)
16	15	bj-imn3ani 34034	. . . . . 6 ⊢ ((ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩) → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
17	9, 14, 16	sylancr 590	. . . . 5 ⊢ (𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
18		opprc1 4789	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, ℂ⟩ = ∅)
19		0ncn 10544	. . . . . . 7 ⊢ ¬ ∅ ∈ ℂ
20		eleq1 2877	. . . . . . 7 ⊢ (⟨𝐴, ℂ⟩ = ∅ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ ∅ ∈ ℂ))
21	19, 20	mtbiri 330	. . . . . 6 ⊢ (⟨𝐴, ℂ⟩ = ∅ → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
22	18, 21	syl 17	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
23	17, 22	pm2.61i 185	. . . 4 ⊢ ¬ ⟨𝐴, ℂ⟩ ∈ ℂ
24		eqcom 2805	. . . . . 6 ⊢ ((+∞ei‘𝐴) = ⟨𝐴, ℂ⟩ ↔ ⟨𝐴, ℂ⟩ = (+∞ei‘𝐴))
25	24	biimpi 219	. . . . 5 ⊢ ((+∞ei‘𝐴) = ⟨𝐴, ℂ⟩ → ⟨𝐴, ℂ⟩ = (+∞ei‘𝐴))
26	25	eleq1d 2874	. . . 4 ⊢ ((+∞ei‘𝐴) = ⟨𝐴, ℂ⟩ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ (+∞ei‘𝐴) ∈ ℂ))
27	23, 26	mtbii 329	. . 3 ⊢ ((+∞ei‘𝐴) = ⟨𝐴, ℂ⟩ → ¬ (+∞ei‘𝐴) ∈ ℂ)
28	7, 27	syl 17	. 2 ⊢ (𝐴 ∈ dom +∞ei → ¬ (+∞ei‘𝐴) ∈ ℂ)
29		ndmfv 6675	. . . 4 ⊢ (¬ 𝐴 ∈ dom +∞ei → (+∞ei‘𝐴) = ∅)
30	29	eleq1d 2874	. . 3 ⊢ (¬ 𝐴 ∈ dom +∞ei → ((+∞ei‘𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
31	19, 30	mtbiri 330	. 2 ⊢ (¬ 𝐴 ∈ dom +∞ei → ¬ (+∞ei‘𝐴) ∈ ℂ)
32	28, 31	pm2.61i 185	1 ⊢ ¬ (+∞ei‘𝐴) ∈ ℂ

Syntax hints:  ¬ wn 3   ↔ wb 209   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∅c0 4243  {csn 4525  {cpr 4527  ⟨cop 4531  dom cdm 5519  ‘cfv 6324  (class class class)co 7135  ℂcc 10524  -cneg 10860  (,]cioc 12727  πcpi 15412  +∞eⁱcinftyexpi 34621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-reg 9040  ax-cnex 10582

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-c 10532  df-bj-inftyexpi 34622

This theorem is referenced by:  bj-ccinftydisj  34628  bj-pinftynrr  34637  bj-minftynrr  34641