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Theorem bj-inftyexpitaudisj 37207
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 6910 . . . . . 6 (𝑥 = 𝐴 → ({R‘(1st𝑥)) = ({R‘(1st𝐴)))
21opeq1d 4878 . . . . 5 (𝑥 = 𝐴 → ⟨({R‘(1st𝑥)), {R}⟩ = ⟨({R‘(1st𝐴)), {R}⟩)
3 df-bj-inftyexpitau 37201 . . . . 5 +∞e = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st𝑥)), {R}⟩)
4 opex 5468 . . . . 5 ⟨({R‘(1st𝐴)), {R}⟩ ∈ V
52, 3, 4fvmpt 7015 . . . 4 (𝐴 ∈ ℝ → (+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩)
6 opex 5468 . . . . 5 ⟨({R‘(1st𝑦)), {R}⟩ ∈ V
7 df-bj-inftyexpitau 37201 . . . . 5 +∞e = (𝑦 ∈ ℝ ↦ ⟨({R‘(1st𝑦)), {R}⟩)
86, 7dmmpti 6711 . . . 4 dom +∞e = ℝ
95, 8eleq2s 2858 . . 3 (𝐴 ∈ dom +∞e → (+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩)
10 nrex1 11105 . . . . . . . 8 R ∈ V
11 bj-nsnid 37072 . . . . . . . 8 (R ∈ V → ¬ {R} ∈ R)
1210, 11ax-mp 5 . . . . . . 7 ¬ {R} ∈ R
1312intnan 486 . . . . . 6 ¬ (({R‘(1st𝐴)) ∈ R ∧ {R} ∈ R)
14 opelxp 5720 . . . . . 6 (⟨({R‘(1st𝐴)), {R}⟩ ∈ (R × R) ↔ (({R‘(1st𝐴)) ∈ R ∧ {R} ∈ R))
1513, 14mtbir 323 . . . . 5 ¬ ⟨({R‘(1st𝐴)), {R}⟩ ∈ (R × R)
16 df-c 11162 . . . . . 6 ℂ = (R × R)
1716eleq2i 2832 . . . . 5 (⟨({R‘(1st𝐴)), {R}⟩ ∈ ℂ ↔ ⟨({R‘(1st𝐴)), {R}⟩ ∈ (R × R))
1815, 17mtbir 323 . . . 4 ¬ ⟨({R‘(1st𝐴)), {R}⟩ ∈ ℂ
19 eqcom 2743 . . . . . 6 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ ↔ ⟨({R‘(1st𝐴)), {R}⟩ = (+∞e𝐴))
2019biimpi 216 . . . . 5 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ → ⟨({R‘(1st𝐴)), {R}⟩ = (+∞e𝐴))
2120eleq1d 2825 . . . 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 11174 . . 3 ¬ ∅ ∈ ℂ
25 ndmfv 6940 . . . 4 𝐴 ∈ dom +∞e → (+∞e𝐴) = ∅)
2625eleq1d 2825 . . 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 1539  wcel 2107  Vcvv 3479  c0 4332  {csn 4625  cop 4631   × cxp 5682  dom cdm 5684  cfv 6560  1st c1st 8013  Rcnr 10906  cc 11154  cr 11155  {Rcfractemp 37198  +∞ecinftyexpitau 37200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-reg 9633  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-omul 8512  df-er 8746  df-ec 8748  df-qs 8752  df-ni 10913  df-pli 10914  df-mi 10915  df-lti 10916  df-plpq 10949  df-mpq 10950  df-ltpq 10951  df-enq 10952  df-nq 10953  df-erq 10954  df-plq 10955  df-mq 10956  df-1nq 10957  df-rq 10958  df-ltnq 10959  df-np 11022  df-plp 11024  df-ltp 11026  df-enr 11096  df-nr 11097  df-c 11162  df-bj-inftyexpitau 37201
This theorem is referenced by: (None)
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