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Theorem bj-inftyexpitaudisj 35607
Description: An element of the circle at infinity is not a complex number. (Contributed by BJ, 4-Feb-2023.)
Assertion
Ref Expression
bj-inftyexpitaudisj ¬ (+∞e𝐴) ∈ ℂ

Proof of Theorem bj-inftyexpitaudisj
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 2fveq3 6844 . . . . . 6 (𝑥 = 𝐴 → ({R‘(1st𝑥)) = ({R‘(1st𝐴)))
21opeq1d 4834 . . . . 5 (𝑥 = 𝐴 → ⟨({R‘(1st𝑥)), {R}⟩ = ⟨({R‘(1st𝐴)), {R}⟩)
3 df-bj-inftyexpitau 35601 . . . . 5 +∞e = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st𝑥)), {R}⟩)
4 opex 5419 . . . . 5 ⟨({R‘(1st𝐴)), {R}⟩ ∈ V
52, 3, 4fvmpt 6945 . . . 4 (𝐴 ∈ ℝ → (+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩)
6 opex 5419 . . . . 5 ⟨({R‘(1st𝑦)), {R}⟩ ∈ V
7 df-bj-inftyexpitau 35601 . . . . 5 +∞e = (𝑦 ∈ ℝ ↦ ⟨({R‘(1st𝑦)), {R}⟩)
86, 7dmmpti 6642 . . . 4 dom +∞e = ℝ
95, 8eleq2s 2856 . . 3 (𝐴 ∈ dom +∞e → (+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩)
10 nrex1 10958 . . . . . . . 8 R ∈ V
11 bj-nsnid 35472 . . . . . . . 8 (R ∈ V → ¬ {R} ∈ R)
1210, 11ax-mp 5 . . . . . . 7 ¬ {R} ∈ R
1312intnan 487 . . . . . 6 ¬ (({R‘(1st𝐴)) ∈ R ∧ {R} ∈ R)
14 opelxp 5667 . . . . . 6 (⟨({R‘(1st𝐴)), {R}⟩ ∈ (R × R) ↔ (({R‘(1st𝐴)) ∈ R ∧ {R} ∈ R))
1513, 14mtbir 322 . . . . 5 ¬ ⟨({R‘(1st𝐴)), {R}⟩ ∈ (R × R)
16 df-c 11015 . . . . . 6 ℂ = (R × R)
1716eleq2i 2829 . . . . 5 (⟨({R‘(1st𝐴)), {R}⟩ ∈ ℂ ↔ ⟨({R‘(1st𝐴)), {R}⟩ ∈ (R × R))
1815, 17mtbir 322 . . . 4 ¬ ⟨({R‘(1st𝐴)), {R}⟩ ∈ ℂ
19 eqcom 2744 . . . . . 6 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ ↔ ⟨({R‘(1st𝐴)), {R}⟩ = (+∞e𝐴))
2019biimpi 215 . . . . 5 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ → ⟨({R‘(1st𝐴)), {R}⟩ = (+∞e𝐴))
2120eleq1d 2822 . . . 4 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ → (⟨({R‘(1st𝐴)), {R}⟩ ∈ ℂ ↔ (+∞e𝐴) ∈ ℂ))
2218, 21mtbii 325 . . 3 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ → ¬ (+∞e𝐴) ∈ ℂ)
239, 22syl 17 . 2 (𝐴 ∈ dom +∞e → ¬ (+∞e𝐴) ∈ ℂ)
24 0ncn 11027 . . 3 ¬ ∅ ∈ ℂ
25 ndmfv 6874 . . . 4 𝐴 ∈ dom +∞e → (+∞e𝐴) = ∅)
2625eleq1d 2822 . . 3 𝐴 ∈ dom +∞e → ((+∞e𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
2724, 26mtbiri 326 . 2 𝐴 ∈ dom +∞e → ¬ (+∞e𝐴) ∈ ℂ)
2823, 27pm2.61i 182 1 ¬ (+∞e𝐴) ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1541  wcel 2106  Vcvv 3443  c0 4280  {csn 4584  cop 4590   × cxp 5629  dom cdm 5631  cfv 6493  1st c1st 7911  Rcnr 10759  cc 11007  cr 11008  {Rcfractemp 35598  +∞ecinftyexpitau 35600
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-reg 9486  ax-inf2 9535
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-1st 7913  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-1o 8404  df-oadd 8408  df-omul 8409  df-er 8606  df-ec 8608  df-qs 8612  df-ni 10766  df-pli 10767  df-mi 10768  df-lti 10769  df-plpq 10802  df-mpq 10803  df-ltpq 10804  df-enq 10805  df-nq 10806  df-erq 10807  df-plq 10808  df-mq 10809  df-1nq 10810  df-rq 10811  df-ltnq 10812  df-np 10875  df-plp 10877  df-ltp 10879  df-enr 10949  df-nr 10950  df-c 11015  df-bj-inftyexpitau 35601
This theorem is referenced by: (None)
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