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Theorem bj-inftyexpidisj 32274
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 4340 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2 df-bj-inftyexpi 32271 . . . . 5 inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3 opex 4859 . . . . 5 𝐴, ℂ⟩ ∈ V
41, 2, 3fvmpt 6191 . . . 4 (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
5 opex 4859 . . . . 5 𝑥, ℂ⟩ ∈ V
65, 2dmmpti 5936 . . . 4 dom inftyexpi = (-π(,]π)
74, 6eleq2s 2706 . . 3 (𝐴 ∈ dom inftyexpi → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
8 cnex 9896 . . . . . . 7 ℂ ∈ V
98prid2 4242 . . . . . 6 ℂ ∈ {𝐴, ℂ}
10 eqid 2610 . . . . . . . 8 {𝐴, ℂ} = {𝐴, ℂ}
1110olci 405 . . . . . . 7 ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})
12 elopg 4861 . . . . . . . 8 ((𝐴 ∈ V ∧ ℂ ∈ V) → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
138, 12mpan2 703 . . . . . . 7 (𝐴 ∈ V → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
1411, 13mpbiri 247 . . . . . 6 (𝐴 ∈ V → {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩)
15 en3lp 8396 . . . . . . 7 ¬ (ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ∧ ⟨𝐴, ℂ⟩ ∈ ℂ)
1615bj-imn3ani 31745 . . . . . 6 ((ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩) → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
179, 14, 16sylancr 694 . . . . 5 (𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
18 opprc1 4363 . . . . . 6 𝐴 ∈ V → ⟨𝐴, ℂ⟩ = ∅)
19 0ncn 9833 . . . . . . 7 ¬ ∅ ∈ ℂ
20 eleq1 2676 . . . . . . 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 175 . . . 4 ¬ ⟨𝐴, ℂ⟩ ∈ ℂ
24 eqcom 2617 . . . . . 6 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ ↔ ⟨𝐴, ℂ⟩ = (inftyexpi ‘𝐴))
2524biimpi 205 . . . . 5 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → ⟨𝐴, ℂ⟩ = (inftyexpi ‘𝐴))
2625eleq1d 2672 . . . 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 6128 . . . 4 𝐴 ∈ dom inftyexpi → (inftyexpi ‘𝐴) = ∅)
3029eleq1d 2672 . . 3 𝐴 ∈ dom inftyexpi → ((inftyexpi ‘𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
3119, 30mtbiri 316 . 2 𝐴 ∈ dom inftyexpi → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
3228, 31pm2.61i 175 1 ¬ (inftyexpi ‘𝐴) ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  wo 382   = wceq 1475  wcel 1977  Vcvv 3173  c0 3874  {csn 4125  {cpr 4127  cop 4131  dom cdm 5038  cfv 5804  (class class class)co 6549  cc 9813  -cneg 10146  (,]cioc 12047  πcpi 14636  inftyexpi cinftyexpi 32270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-cnex 9871
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-c 9821  df-bj-inftyexpi 32271
This theorem is referenced by:  bj-ccinftydisj  32277  bj-pinftynrr  32286  bj-minftynrr  32290
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