Theorem bj-inftyexpidisj 37459

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 4831	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2		df-bj-inftyexpi 37456	. . . . 5 ⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3		opex 5419	. . . . 5 ⊢ ⟨𝐴, ℂ⟩ ∈ V
4	1, 2, 3	fvmpt 6949	. . . 4 ⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) = ⟨𝐴, ℂ⟩)
5		opex 5419	. . . . 5 ⊢ ⟨𝑥, ℂ⟩ ∈ V
6	5, 2	dmmpti 6644	. . . 4 ⊢ dom +∞ei = (-π(,]π)
7	4, 6	eleq2s 2855	. . 3 ⊢ (𝐴 ∈ dom +∞ei → (+∞ei‘𝐴) = ⟨𝐴, ℂ⟩)
8		cnex 11119	. . . . . . 7 ⊢ ℂ ∈ V
9	8	prid2 4722	. . . . . 6 ⊢ ℂ ∈ {𝐴, ℂ}
10		eqid 2737	. . . . . . . 8 ⊢ {𝐴, ℂ} = {𝐴, ℂ}
11	10	olci 867	. . . . . . 7 ⊢ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})
12		elopg 5422	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ℂ ∈ V) → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
13	8, 12	mpan2 692	. . . . . . 7 ⊢ (𝐴 ∈ V → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
14	11, 13	mpbiri 258	. . . . . 6 ⊢ (𝐴 ∈ V → {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩)
15		en3lp 9535	. . . . . . 7 ⊢ ¬ (ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ∧ ⟨𝐴, ℂ⟩ ∈ ℂ)
16	15	bj-imn3ani 36808	. . . . . 6 ⊢ ((ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩) → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
17	9, 14, 16	sylancr 588	. . . . 5 ⊢ (𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
18		opprc1 4855	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, ℂ⟩ = ∅)
19		0ncn 11056	. . . . . . 7 ⊢ ¬ ∅ ∈ ℂ
20		eleq1 2825	. . . . . . 7 ⊢ (⟨𝐴, ℂ⟩ = ∅ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ ∅ ∈ ℂ))
21	19, 20	mtbiri 327	. . . . . 6 ⊢ (⟨𝐴, ℂ⟩ = ∅ → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
22	18, 21	syl 17	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
23	17, 22	pm2.61i 182	. . . 4 ⊢ ¬ ⟨𝐴, ℂ⟩ ∈ ℂ
24		eqcom 2744	. . . . . 6 ⊢ ((+∞ei‘𝐴) = ⟨𝐴, ℂ⟩ ↔ ⟨𝐴, ℂ⟩ = (+∞ei‘𝐴))
25	24	biimpi 216	. . . . 5 ⊢ ((+∞ei‘𝐴) = ⟨𝐴, ℂ⟩ → ⟨𝐴, ℂ⟩ = (+∞ei‘𝐴))
26	25	eleq1d 2822	. . . 4 ⊢ ((+∞ei‘𝐴) = ⟨𝐴, ℂ⟩ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ (+∞ei‘𝐴) ∈ ℂ))
27	23, 26	mtbii 326	. . 3 ⊢ ((+∞ei‘𝐴) = ⟨𝐴, ℂ⟩ → ¬ (+∞ei‘𝐴) ∈ ℂ)
28	7, 27	syl 17	. 2 ⊢ (𝐴 ∈ dom +∞ei → ¬ (+∞ei‘𝐴) ∈ ℂ)
29		ndmfv 6874	. . . 4 ⊢ (¬ 𝐴 ∈ dom +∞ei → (+∞ei‘𝐴) = ∅)
30	29	eleq1d 2822	. . 3 ⊢ (¬ 𝐴 ∈ dom +∞ei → ((+∞ei‘𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
31	19, 30	mtbiri 327	. 2 ⊢ (¬ 𝐴 ∈ dom +∞ei → ¬ (+∞ei‘𝐴) ∈ ℂ)
32	28, 31	pm2.61i 182	1 ⊢ ¬ (+∞ei‘𝐴) ∈ ℂ

Syntax hints:  ¬ wn 3   ↔ wb 206   ∨ wo 848   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  {csn 4582  {cpr 4584  ⟨cop 4588  dom cdm 5632  ‘cfv 6500  (class class class)co 7368  ℂcc 11036  -cneg 11377  (,]cioc 13274  πcpi 16001  +∞eⁱcinftyexpi 37455

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690  ax-reg 9509  ax-cnex 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508  df-c 11044  df-bj-inftyexpi 37456

This theorem is referenced by:  bj-ccinftydisj  37462  bj-pinftynrr  37471  bj-minftynrr  37475