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Theorem bj-inftyexpitaudisj 37188
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 6912 . . . . . 6 (𝑥 = 𝐴 → ({R‘(1st𝑥)) = ({R‘(1st𝐴)))
21opeq1d 4884 . . . . 5 (𝑥 = 𝐴 → ⟨({R‘(1st𝑥)), {R}⟩ = ⟨({R‘(1st𝐴)), {R}⟩)
3 df-bj-inftyexpitau 37182 . . . . 5 +∞e = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st𝑥)), {R}⟩)
4 opex 5475 . . . . 5 ⟨({R‘(1st𝐴)), {R}⟩ ∈ V
52, 3, 4fvmpt 7016 . . . 4 (𝐴 ∈ ℝ → (+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩)
6 opex 5475 . . . . 5 ⟨({R‘(1st𝑦)), {R}⟩ ∈ V
7 df-bj-inftyexpitau 37182 . . . . 5 +∞e = (𝑦 ∈ ℝ ↦ ⟨({R‘(1st𝑦)), {R}⟩)
86, 7dmmpti 6713 . . . 4 dom +∞e = ℝ
95, 8eleq2s 2857 . . 3 (𝐴 ∈ dom +∞e → (+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩)
10 nrex1 11102 . . . . . . . 8 R ∈ V
11 bj-nsnid 37053 . . . . . . . 8 (R ∈ V → ¬ {R} ∈ R)
1210, 11ax-mp 5 . . . . . . 7 ¬ {R} ∈ R
1312intnan 486 . . . . . 6 ¬ (({R‘(1st𝐴)) ∈ R ∧ {R} ∈ R)
14 opelxp 5725 . . . . . 6 (⟨({R‘(1st𝐴)), {R}⟩ ∈ (R × R) ↔ (({R‘(1st𝐴)) ∈ R ∧ {R} ∈ R))
1513, 14mtbir 323 . . . . 5 ¬ ⟨({R‘(1st𝐴)), {R}⟩ ∈ (R × R)
16 df-c 11159 . . . . . 6 ℂ = (R × R)
1716eleq2i 2831 . . . . 5 (⟨({R‘(1st𝐴)), {R}⟩ ∈ ℂ ↔ ⟨({R‘(1st𝐴)), {R}⟩ ∈ (R × R))
1815, 17mtbir 323 . . . 4 ¬ ⟨({R‘(1st𝐴)), {R}⟩ ∈ ℂ
19 eqcom 2742 . . . . . 6 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ ↔ ⟨({R‘(1st𝐴)), {R}⟩ = (+∞e𝐴))
2019biimpi 216 . . . . 5 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ → ⟨({R‘(1st𝐴)), {R}⟩ = (+∞e𝐴))
2120eleq1d 2824 . . . 4 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ → (⟨({R‘(1st𝐴)), {R}⟩ ∈ ℂ ↔ (+∞e𝐴) ∈ ℂ))
2218, 21mtbii 326 . . 3 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ → ¬ (+∞e𝐴) ∈ ℂ)
239, 22syl 17 . 2 (𝐴 ∈ dom +∞e → ¬ (+∞e𝐴) ∈ ℂ)
24 0ncn 11171 . . 3 ¬ ∅ ∈ ℂ
25 ndmfv 6942 . . . 4 𝐴 ∈ dom +∞e → (+∞e𝐴) = ∅)
2625eleq1d 2824 . . 3 𝐴 ∈ dom +∞e → ((+∞e𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
2724, 26mtbiri 327 . 2 𝐴 ∈ dom +∞e → ¬ (+∞e𝐴) ∈ ℂ)
2823, 27pm2.61i 182 1 ¬ (+∞e𝐴) ∈ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  {csn 4631  cop 4637   × cxp 5687  dom cdm 5689  cfv 6563  1st c1st 8011  Rcnr 10903  cc 11151  cr 11152  {Rcfractemp 37179  +∞ecinftyexpitau 37181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-ec 8746  df-qs 8750  df-ni 10910  df-pli 10911  df-mi 10912  df-lti 10913  df-plpq 10946  df-mpq 10947  df-ltpq 10948  df-enq 10949  df-nq 10950  df-erq 10951  df-plq 10952  df-mq 10953  df-1nq 10954  df-rq 10955  df-ltnq 10956  df-np 11019  df-plp 11021  df-ltp 11023  df-enr 11093  df-nr 11094  df-c 11159  df-bj-inftyexpitau 37182
This theorem is referenced by: (None)
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