Theorem bj-inftyexpidisj 33097

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 4402	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2		df-bj-inftyexpi 33094	. . . . 5 ⊢ inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3		opex 4932	. . . . 5 ⊢ ⟨𝐴, ℂ⟩ ∈ V
4	1, 2, 3	fvmpt 6282	. . . 4 ⊢ (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
5		opex 4932	. . . . 5 ⊢ ⟨𝑥, ℂ⟩ ∈ V
6	5, 2	dmmpti 6023	. . . 4 ⊢ dom inftyexpi = (-π(,]π)
7	4, 6	eleq2s 2719	. . 3 ⊢ (𝐴 ∈ dom inftyexpi → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
8		cnex 10017	. . . . . . 7 ⊢ ℂ ∈ V
9	8	prid2 4298	. . . . . 6 ⊢ ℂ ∈ {𝐴, ℂ}
10		eqid 2622	. . . . . . . 8 ⊢ {𝐴, ℂ} = {𝐴, ℂ}
11	10	olci 406	. . . . . . 7 ⊢ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})
12		elopg 4934	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ℂ ∈ V) → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
13	8, 12	mpan2 707	. . . . . . 7 ⊢ (𝐴 ∈ V → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
14	11, 13	mpbiri 248	. . . . . 6 ⊢ (𝐴 ∈ V → {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩)
15		en3lp 8513	. . . . . . 7 ⊢ ¬ (ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ∧ ⟨𝐴, ℂ⟩ ∈ ℂ)
16	15	bj-imn3ani 32572	. . . . . 6 ⊢ ((ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩) → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
17	9, 14, 16	sylancr 695	. . . . 5 ⊢ (𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
18		opprc1 4425	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, ℂ⟩ = ∅)
19		0ncn 9954	. . . . . . 7 ⊢ ¬ ∅ ∈ ℂ
20		eleq1 2689	. . . . . . 7 ⊢ (⟨𝐴, ℂ⟩ = ∅ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ ∅ ∈ ℂ))
21	19, 20	mtbiri 317	. . . . . 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 2629	. . . . . 6 ⊢ ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ ↔ ⟨𝐴, ℂ⟩ = (inftyexpi ‘𝐴))
25	24	biimpi 206	. . . . 5 ⊢ ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → ⟨𝐴, ℂ⟩ = (inftyexpi ‘𝐴))
26	25	eleq1d 2686	. . . 4 ⊢ ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ (inftyexpi ‘𝐴) ∈ ℂ))
27	23, 26	mtbii 316	. . 3 ⊢ ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
28	7, 27	syl 17	. 2 ⊢ (𝐴 ∈ dom inftyexpi → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
29		ndmfv 6218	. . . 4 ⊢ (¬ 𝐴 ∈ dom inftyexpi → (inftyexpi ‘𝐴) = ∅)
30	29	eleq1d 2686	. . 3 ⊢ (¬ 𝐴 ∈ dom inftyexpi → ((inftyexpi ‘𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
31	19, 30	mtbiri 317	. 2 ⊢ (¬ 𝐴 ∈ dom inftyexpi → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
32	28, 31	pm2.61i 176	1 ⊢ ¬ (inftyexpi ‘𝐴) ∈ ℂ

Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 383   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  {csn 4177  {cpr 4179  ⟨cop 4183  dom cdm 5114  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  -cneg 10267  (,]cioc 12176  πcpi 14797  inftyexpi cinftyexpi 33093

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-cnex 9992

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  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-tp 4182  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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-c 9942  df-bj-inftyexpi 33094

This theorem is referenced by:  bj-ccinftydisj  33100  bj-pinftynrr  33109  bj-minftynrr  33113