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Theorem bj-inftyexpidisj 33227
 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.
StepHypRef Expression
1 opeq1 4433 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2 df-bj-inftyexpi 33224 . . . . 5 inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3 opex 4962 . . . . 5 𝐴, ℂ⟩ ∈ V
41, 2, 3fvmpt 6321 . . . 4 (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
5 opex 4962 . . . . 5 𝑥, ℂ⟩ ∈ V
65, 2dmmpti 6061 . . . 4 dom inftyexpi = (-π(,]π)
74, 6eleq2s 2748 . . 3 (𝐴 ∈ dom inftyexpi → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
8 cnex 10055 . . . . . . 7 ℂ ∈ V
98prid2 4330 . . . . . 6 ℂ ∈ {𝐴, ℂ}
10 eqid 2651 . . . . . . . 8 {𝐴, ℂ} = {𝐴, ℂ}
1110olci 405 . . . . . . 7 ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})
12 elopg 4964 . . . . . . . 8 ((𝐴 ∈ V ∧ ℂ ∈ V) → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
138, 12mpan2 707 . . . . . . 7 (𝐴 ∈ V → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
1411, 13mpbiri 248 . . . . . 6 (𝐴 ∈ V → {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩)
15 en3lp 8551 . . . . . . 7 ¬ (ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ∧ ⟨𝐴, ℂ⟩ ∈ ℂ)
1615bj-imn3ani 32697 . . . . . 6 ((ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩) → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
179, 14, 16sylancr 696 . . . . 5 (𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
18 opprc1 4457 . . . . . 6 𝐴 ∈ V → ⟨𝐴, ℂ⟩ = ∅)
19 0ncn 9992 . . . . . . 7 ¬ ∅ ∈ ℂ
20 eleq1 2718 . . . . . . 7 (⟨𝐴, ℂ⟩ = ∅ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ ∅ ∈ ℂ))
2119, 20mtbiri 316 . . . . . 6 (⟨𝐴, ℂ⟩ = ∅ → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
2218, 21syl 17 . . . . 5 𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
2317, 22pm2.61i 176 . . . 4 ¬ ⟨𝐴, ℂ⟩ ∈ ℂ
24 eqcom 2658 . . . . . 6 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ ↔ ⟨𝐴, ℂ⟩ = (inftyexpi ‘𝐴))
2524biimpi 206 . . . . 5 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → ⟨𝐴, ℂ⟩ = (inftyexpi ‘𝐴))
2625eleq1d 2715 . . . 4 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ (inftyexpi ‘𝐴) ∈ ℂ))
2723, 26mtbii 315 . . 3 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
287, 27syl 17 . 2 (𝐴 ∈ dom inftyexpi → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
29 ndmfv 6256 . . . 4 𝐴 ∈ dom inftyexpi → (inftyexpi ‘𝐴) = ∅)
3029eleq1d 2715 . . 3 𝐴 ∈ dom inftyexpi → ((inftyexpi ‘𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
3119, 30mtbiri 316 . 2 𝐴 ∈ dom inftyexpi → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
3228, 31pm2.61i 176 1 ¬ (inftyexpi ‘𝐴) ∈ ℂ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 382   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ∅c0 3948  {csn 4210  {cpr 4212  ⟨cop 4216  dom cdm 5143  ‘cfv 5926  (class class class)co 6690  ℂcc 9972  -cneg 10305  (,]cioc 12214  πcpi 14841  inftyexpi cinftyexpi 33223 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-cnex 10030 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-c 9980  df-bj-inftyexpi 33224 This theorem is referenced by:  bj-ccinftydisj  33230  bj-pinftynrr  33239  bj-minftynrr  33243
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