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Theorem bj-inftyexpidisj 34538
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 4787 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2 df-bj-inftyexpi 34535 . . . . 5 +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3 opex 5343 . . . . 5 𝐴, ℂ⟩ ∈ V
41, 2, 3fvmpt 6756 . . . 4 (𝐴 ∈ (-π(,]π) → (+∞ei𝐴) = ⟨𝐴, ℂ⟩)
5 opex 5343 . . . . 5 𝑥, ℂ⟩ ∈ V
65, 2dmmpti 6480 . . . 4 dom +∞ei = (-π(,]π)
74, 6eleq2s 2934 . . 3 (𝐴 ∈ dom +∞ei → (+∞ei𝐴) = ⟨𝐴, ℂ⟩)
8 cnex 10610 . . . . . . 7 ℂ ∈ V
98prid2 4683 . . . . . 6 ℂ ∈ {𝐴, ℂ}
10 eqid 2824 . . . . . . . 8 {𝐴, ℂ} = {𝐴, ℂ}
1110olci 863 . . . . . . 7 ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})
12 elopg 5345 . . . . . . . 8 ((𝐴 ∈ V ∧ ℂ ∈ V) → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
138, 12mpan2 690 . . . . . . 7 (𝐴 ∈ V → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
1411, 13mpbiri 261 . . . . . 6 (𝐴 ∈ V → {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩)
15 en3lp 9068 . . . . . . 7 ¬ (ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ∧ ⟨𝐴, ℂ⟩ ∈ ℂ)
1615bj-imn3ani 33946 . . . . . 6 ((ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩) → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
179, 14, 16sylancr 590 . . . . 5 (𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
18 opprc1 4813 . . . . . 6 𝐴 ∈ V → ⟨𝐴, ℂ⟩ = ∅)
19 0ncn 10547 . . . . . . 7 ¬ ∅ ∈ ℂ
20 eleq1 2903 . . . . . . 7 (⟨𝐴, ℂ⟩ = ∅ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ ∅ ∈ ℂ))
2119, 20mtbiri 330 . . . . . 6 (⟨𝐴, ℂ⟩ = ∅ → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
2218, 21syl 17 . . . . 5 𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
2317, 22pm2.61i 185 . . . 4 ¬ ⟨𝐴, ℂ⟩ ∈ ℂ
24 eqcom 2831 . . . . . 6 ((+∞ei𝐴) = ⟨𝐴, ℂ⟩ ↔ ⟨𝐴, ℂ⟩ = (+∞ei𝐴))
2524biimpi 219 . . . . 5 ((+∞ei𝐴) = ⟨𝐴, ℂ⟩ → ⟨𝐴, ℂ⟩ = (+∞ei𝐴))
2625eleq1d 2900 . . . 4 ((+∞ei𝐴) = ⟨𝐴, ℂ⟩ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ (+∞ei𝐴) ∈ ℂ))
2723, 26mtbii 329 . . 3 ((+∞ei𝐴) = ⟨𝐴, ℂ⟩ → ¬ (+∞ei𝐴) ∈ ℂ)
287, 27syl 17 . 2 (𝐴 ∈ dom +∞ei → ¬ (+∞ei𝐴) ∈ ℂ)
29 ndmfv 6688 . . . 4 𝐴 ∈ dom +∞ei → (+∞ei𝐴) = ∅)
3029eleq1d 2900 . . 3 𝐴 ∈ dom +∞ei → ((+∞ei𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
3119, 30mtbiri 330 . 2 𝐴 ∈ dom +∞ei → ¬ (+∞ei𝐴) ∈ ℂ)
3228, 31pm2.61i 185 1 ¬ (+∞ei𝐴) ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wo 844   = wceq 1538  wcel 2115  Vcvv 3480  c0 4275  {csn 4549  {cpr 4551  cop 4555  dom cdm 5542  cfv 6343  (class class class)co 7145  cc 10527  -cneg 10863  (,]cioc 12732  πcpi 15416  +∞eicinftyexpi 34534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-reg 9047  ax-cnex 10585
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-c 10535  df-bj-inftyexpi 34535
This theorem is referenced by:  bj-ccinftydisj  34541  bj-pinftynrr  34550  bj-minftynrr  34554
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