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Theorem bj-inftyexpidisj 33531
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 4559 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2 df-bj-inftyexpi 33528 . . . . 5 inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3 opex 5088 . . . . 5 𝐴, ℂ⟩ ∈ V
41, 2, 3fvmpt 6471 . . . 4 (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
5 opex 5088 . . . . 5 𝑥, ℂ⟩ ∈ V
65, 2dmmpti 6201 . . . 4 dom inftyexpi = (-π(,]π)
74, 6eleq2s 2862 . . 3 (𝐴 ∈ dom inftyexpi → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
8 cnex 10270 . . . . . . 7 ℂ ∈ V
98prid2 4453 . . . . . 6 ℂ ∈ {𝐴, ℂ}
10 eqid 2765 . . . . . . . 8 {𝐴, ℂ} = {𝐴, ℂ}
1110olci 892 . . . . . . 7 ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})
12 elopg 5090 . . . . . . . 8 ((𝐴 ∈ V ∧ ℂ ∈ V) → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
138, 12mpan2 682 . . . . . . 7 (𝐴 ∈ V → ({𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ↔ ({𝐴, ℂ} = {𝐴} ∨ {𝐴, ℂ} = {𝐴, ℂ})))
1411, 13mpbiri 249 . . . . . 6 (𝐴 ∈ V → {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩)
15 en3lp 8724 . . . . . . 7 ¬ (ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩ ∧ ⟨𝐴, ℂ⟩ ∈ ℂ)
1615bj-imn3ani 33008 . . . . . 6 ((ℂ ∈ {𝐴, ℂ} ∧ {𝐴, ℂ} ∈ ⟨𝐴, ℂ⟩) → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
179, 14, 16sylancr 581 . . . . 5 (𝐴 ∈ V → ¬ ⟨𝐴, ℂ⟩ ∈ ℂ)
18 opprc1 4583 . . . . . 6 𝐴 ∈ V → ⟨𝐴, ℂ⟩ = ∅)
19 0ncn 10207 . . . . . . 7 ¬ ∅ ∈ ℂ
20 eleq1 2832 . . . . . . 7 (⟨𝐴, ℂ⟩ = ∅ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ ∅ ∈ ℂ))
2119, 20mtbiri 318 . . . . . 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 2772 . . . . . 6 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ ↔ ⟨𝐴, ℂ⟩ = (inftyexpi ‘𝐴))
2524biimpi 207 . . . . 5 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → ⟨𝐴, ℂ⟩ = (inftyexpi ‘𝐴))
2625eleq1d 2829 . . . 4 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → (⟨𝐴, ℂ⟩ ∈ ℂ ↔ (inftyexpi ‘𝐴) ∈ ℂ))
2723, 26mtbii 317 . . 3 ((inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩ → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
287, 27syl 17 . 2 (𝐴 ∈ dom inftyexpi → ¬ (inftyexpi ‘𝐴) ∈ ℂ)
29 ndmfv 6405 . . . 4 𝐴 ∈ dom inftyexpi → (inftyexpi ‘𝐴) = ∅)
3029eleq1d 2829 . . 3 𝐴 ∈ dom inftyexpi → ((inftyexpi ‘𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
3119, 30mtbiri 318 . 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 197  wo 873   = wceq 1652  wcel 2155  Vcvv 3350  c0 4079  {csn 4334  {cpr 4336  cop 4340  dom cdm 5277  cfv 6068  (class class class)co 6842  cc 10187  -cneg 10521  (,]cioc 12378  πcpi 15079  inftyexpi cinftyexpi 33527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-reg 8704  ax-cnex 10245
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fn 6071  df-fv 6076  df-c 10195  df-bj-inftyexpi 33528
This theorem is referenced by:  bj-ccinftydisj  33534  bj-pinftynrr  33543  bj-minftynrr  33547
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