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Theorem bj-inftyexpidisj 36820
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 ¬ (+∞ei𝐴) ∈ ℂ

Proof of Theorem bj-inftyexpidisj
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 opeq1 4875 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2 df-bj-inftyexpi 36817 . . . . 5 +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3 opex 5466 . . . . 5 𝐴, ℂ⟩ ∈ V
41, 2, 3fvmpt 7004 . . . 4 (𝐴 ∈ (-π(,]π) → (+∞ei𝐴) = ⟨𝐴, ℂ⟩)
5 opex 5466 . . . . 5 𝑥, ℂ⟩ ∈ V
65, 2dmmpti 6700 . . . 4 dom +∞ei = (-π(,]π)
74, 6eleq2s 2843 . . 3 (𝐴 ∈ dom +∞ei → (+∞ei𝐴) = ⟨𝐴, ℂ⟩)
8 cnex 11221 . . . . . . 7 ℂ ∈ V
98prid2 4769 . . . . . 6 ℂ ∈ {𝐴, ℂ}
10 eqid 2725 . . . . . . . 8 {𝐴, ℂ} = {𝐴, ℂ}
1110olci 864 . . . . . . 7 ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})
12 elopg 5468 . . . . . . . 8 ((𝐴 ∈ V ∧ ℂ ∈ V) → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
138, 12mpan2 689 . . . . . . 7 (𝐴 ∈ V → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
1411, 13mpbiri 257 . . . . . 6 (𝐴 ∈ V → {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩)
15 en3lp 9639 . . . . . . 7 ¬ (ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ∧ ⟨𝐴, ℂ⟩ ∈ ℂ)
1615bj-imn3ani 36195 . . . . . 6 ((ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩) → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
179, 14, 16sylancr 585 . . . . 5 (𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
18 opprc1 4899 . . . . . 6 𝐴 ∈ V → ⟨𝐴, ℂ⟩ = ∅)
19 0ncn 11158 . . . . . . 7 ¬ ∅ ∈ ℂ
20 eleq1 2813 . . . . . . 7 (⟨𝐴, ℂ⟩ = ∅ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ ∅ ∈ ℂ))
2119, 20mtbiri 326 . . . . . 6 (⟨𝐴, ℂ⟩ = ∅ → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
2218, 21syl 17 . . . . 5 𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
2317, 22pm2.61i 182 . . . 4 ¬ ⟨𝐴, ℂ⟩ ∈ ℂ
24 eqcom 2732 . . . . . 6 ((+∞ei𝐴) = ⟨𝐴, ℂ⟩ ↔ ⟨𝐴, ℂ⟩ = (+∞ei𝐴))
2524biimpi 215 . . . . 5 ((+∞ei𝐴) = ⟨𝐴, ℂ⟩ → ⟨𝐴, ℂ⟩ = (+∞ei𝐴))
2625eleq1d 2810 . . . 4 ((+∞ei𝐴) = ⟨𝐴, ℂ⟩ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ (+∞ei𝐴) ∈ ℂ))
2723, 26mtbii 325 . . 3 ((+∞ei𝐴) = ⟨𝐴, ℂ⟩ → ¬ (+∞ei𝐴) ∈ ℂ)
287, 27syl 17 . 2 (𝐴 ∈ dom +∞ei → ¬ (+∞ei𝐴) ∈ ℂ)
29 ndmfv 6931 . . . 4 𝐴 ∈ dom +∞ei → (+∞ei𝐴) = ∅)
3029eleq1d 2810 . . 3 𝐴 ∈ dom +∞ei → ((+∞ei𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
3119, 30mtbiri 326 . 2 𝐴 ∈ dom +∞ei → ¬ (+∞ei𝐴) ∈ ℂ)
3228, 31pm2.61i 182 1 ¬ (+∞ei𝐴) ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wo 845   = wceq 1533  wcel 2098  Vcvv 3461  c0 4322  {csn 4630  {cpr 4632  cop 4636  dom cdm 5678  cfv 6549  (class class class)co 7419  cc 11138  -cneg 11477  (,]cioc 13360  πcpi 16046  +∞eicinftyexpi 36816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741  ax-reg 9617  ax-cnex 11196
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fn 6552  df-fv 6557  df-c 11146  df-bj-inftyexpi 36817
This theorem is referenced by:  bj-ccinftydisj  36823  bj-pinftynrr  36832  bj-minftynrr  36836
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