Theorem bj-inftyexpidisj 34486

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 4796	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2		df-bj-inftyexpi 34483	. . . . 5 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3		opex 5348	. . . . 5 ⊢ ⟨𝐴, ℂ⟩ ∈ V
4	1, 2, 3	fvmpt 6762	. . . 4 ⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) = ⟨𝐴, ℂ⟩)
5		opex 5348	. . . . 5 ⊢ ⟨𝑥, ℂ⟩ ∈ V
6	5, 2	dmmpti 6486	. . . 4 ⊢ dom +∞ei = (-π(,]π)
7	4, 6	eleq2s 2931	. . 3 ⊢ (𝐴 ∈ dom +∞ei → (+∞ei‘𝐴) = ⟨𝐴, ℂ⟩)
8		cnex 10612	. . . . . . 7 ⊢ ℂ ∈ V
9	8	prid2 4692	. . . . . 6 ⊢ ℂ ∈ {𝐴, ℂ}
10		eqid 2821	. . . . . . . 8 ⊢ {𝐴, ℂ} = {𝐴, ℂ}
11	10	olci 862	. . . . . . 7 ⊢ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})
12		elopg 5350	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ℂ ∈ V) → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
13	8, 12	mpan2 689	. . . . . . 7 ⊢ (𝐴 ∈ V → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
14	11, 13	mpbiri 260	. . . . . 6 ⊢ (𝐴 ∈ V → {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩)
15		en3lp 9071	. . . . . . 7 ⊢ ¬ (ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ∧ ⟨𝐴, ℂ⟩ ∈ ℂ)
16	15	bj-imn3ani 33916	. . . . . 6 ⊢ ((ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩) → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
17	9, 14, 16	sylancr 589	. . . . 5 ⊢ (𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
18		opprc1 4820	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, ℂ⟩ = ∅)
19		0ncn 10549	. . . . . . 7 ⊢ ¬ ∅ ∈ ℂ
20		eleq1 2900	. . . . . . 7 ⊢ (⟨𝐴, ℂ⟩ = ∅ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ ∅ ∈ ℂ))
21	19, 20	mtbiri 329	. . . . . 6 ⊢ (⟨𝐴, ℂ⟩ = ∅ → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
22	18, 21	syl 17	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
23	17, 22	pm2.61i 184	. . . 4 ⊢ ¬ ⟨𝐴, ℂ⟩ ∈ ℂ
24		eqcom 2828	. . . . . 6 ⊢ ((+∞ei‘𝐴) = ⟨𝐴, ℂ⟩ ↔ ⟨𝐴, ℂ⟩ = (+∞ei‘𝐴))
25	24	biimpi 218	. . . . 5 ⊢ ((+∞ei‘𝐴) = ⟨𝐴, ℂ⟩ → ⟨𝐴, ℂ⟩ = (+∞ei‘𝐴))
26	25	eleq1d 2897	. . . 4 ⊢ ((+∞ei‘𝐴) = ⟨𝐴, ℂ⟩ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ (+∞ei‘𝐴) ∈ ℂ))
27	23, 26	mtbii 328	. . 3 ⊢ ((+∞ei‘𝐴) = ⟨𝐴, ℂ⟩ → ¬ (+∞ei‘𝐴) ∈ ℂ)
28	7, 27	syl 17	. 2 ⊢ (𝐴 ∈ dom +∞ei → ¬ (+∞ei‘𝐴) ∈ ℂ)
29		ndmfv 6694	. . . 4 ⊢ (¬ 𝐴 ∈ dom +∞ei → (+∞ei‘𝐴) = ∅)
30	29	eleq1d 2897	. . 3 ⊢ (¬ 𝐴 ∈ dom +∞ei → ((+∞ei‘𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
31	19, 30	mtbiri 329	. 2 ⊢ (¬ 𝐴 ∈ dom +∞ei → ¬ (+∞ei‘𝐴) ∈ ℂ)
32	28, 31	pm2.61i 184	1 ⊢ ¬ (+∞ei‘𝐴) ∈ ℂ

Syntax hints:  ¬ wn 3   ↔ wb 208   ∨ wo 843   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ∅c0 4290  {csn 4560  {cpr 4562  ⟨cop 4566  dom cdm 5549  ‘cfv 6349  (class class class)co 7150  ℂcc 10529  -cneg 10865  (,]cioc 12733  πcpi 15414  +∞eⁱcinftyexpi 34482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-reg 9050  ax-cnex 10587

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357  df-c 10537  df-bj-inftyexpi 34483

This theorem is referenced by:  bj-ccinftydisj  34489  bj-pinftynrr  34498  bj-minftynrr  34502