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Theorem bj-inftyexpidisj 32098
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 4334 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2 df-bj-inftyexpi 32095 . . . . 5 inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3 opex 4853 . . . . 5 𝐴, ℂ⟩ ∈ V
41, 2, 3fvmpt 6176 . . . 4 (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
5 opex 4853 . . . . 5 𝑥, ℂ⟩ ∈ V
65, 2dmmpti 5922 . . . 4 dom inftyexpi = (-π(,]π)
74, 6eleq2s 2705 . . 3 (𝐴 ∈ dom inftyexpi → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
8 cnex 9874 . . . . . . 7 ℂ ∈ V
98prid2 4241 . . . . . 6 ℂ ∈ {𝐴, ℂ}
10 eqid 2609 . . . . . . . 8 {𝐴, ℂ} = {𝐴, ℂ}
1110olci 404 . . . . . . 7 ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})
12 elopg 4855 . . . . . . . 8 ((𝐴 ∈ V ∧ ℂ ∈ V) → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
138, 12mpan2 702 . . . . . . 7 (𝐴 ∈ V → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
1411, 13mpbiri 246 . . . . . 6 (𝐴 ∈ V → {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩)
15 en3lp 8374 . . . . . . 7 ¬ (ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ∧ ⟨𝐴, ℂ⟩ ∈ ℂ)
1615bj-imn3ani 31579 . . . . . 6 ((ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩) → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
179, 14, 16sylancr 693 . . . . 5 (𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
18 opprc1 4357 . . . . . 6 𝐴 ∈ V → ⟨𝐴, ℂ⟩ = ∅)
19 0ncn 9811 . . . . . . 7 ¬ ∅ ∈ ℂ
20 eleq1 2675 . . . . . . 7 (⟨𝐴, ℂ⟩ = ∅ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ ∅ ∈ ℂ))
2119, 20mtbiri 315 . . . . . 6 (⟨𝐴, ℂ⟩ = ∅ → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
2218, 21syl 17 . . . . 5 𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
2317, 22pm2.61i 174 . . . 4 ¬ ⟨𝐴, ℂ⟩ ∈ ℂ
24 eqcom 2616 . . . . . 6 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ ↔ ⟨𝐴, ℂ⟩ = (inftyexpi ‘𝐴))
2524biimpi 204 . . . . 5 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → ⟨𝐴, ℂ⟩ = (inftyexpi ‘𝐴))
2625eleq1d 2671 . . . 4 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ (inftyexpi ‘𝐴) ∈ ℂ))
2723, 26mtbii 314 . . 3 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
287, 27syl 17 . 2 (𝐴 ∈ dom inftyexpi → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
29 ndmfv 6113 . . . 4 𝐴 ∈ dom inftyexpi → (inftyexpi ‘𝐴) = ∅)
3029eleq1d 2671 . . 3 𝐴 ∈ dom inftyexpi → ((inftyexpi ‘𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
3119, 30mtbiri 315 . 2 𝐴 ∈ dom inftyexpi → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
3228, 31pm2.61i 174 1 ¬ (inftyexpi ‘𝐴) ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wo 381   = wceq 1474  wcel 1976  Vcvv 3172  c0 3873  {csn 4124  {cpr 4126  cop 4130  dom cdm 5028  cfv 5790  (class class class)co 6527  cc 9791  -cneg 10119  (,]cioc 12006  πcpi 14585  inftyexpi cinftyexpi 32094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-reg 8358  ax-cnex 9849
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798  df-c 9799  df-bj-inftyexpi 32095
This theorem is referenced by:  bj-ccinftydisj  32101  bj-pinftynrr  32110  bj-minftynrr  32114
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