Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-inftyexpitaudisj Structured version   Visualization version   GIF version

Theorem bj-inftyexpitaudisj 34504
 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 6651 . . . . . 6 (𝑥 = 𝐴 → ({R‘(1st𝑥)) = ({R‘(1st𝐴)))
21opeq1d 4785 . . . . 5 (𝑥 = 𝐴 → ⟨({R‘(1st𝑥)), {R}⟩ = ⟨({R‘(1st𝐴)), {R}⟩)
3 df-bj-inftyexpitau 34498 . . . . 5 +∞e = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st𝑥)), {R}⟩)
4 opex 5332 . . . . 5 ⟨({R‘(1st𝐴)), {R}⟩ ∈ V
52, 3, 4fvmpt 6744 . . . 4 (𝐴 ∈ ℝ → (+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩)
6 opex 5332 . . . . 5 ⟨({R‘(1st𝑦)), {R}⟩ ∈ V
7 df-bj-inftyexpitau 34498 . . . . 5 +∞e = (𝑦 ∈ ℝ ↦ ⟨({R‘(1st𝑦)), {R}⟩)
86, 7dmmpti 6468 . . . 4 dom +∞e = ℝ
95, 8eleq2s 2929 . . 3 (𝐴 ∈ dom +∞e → (+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩)
10 nrex1 10464 . . . . . . . 8 R ∈ V
11 bj-nsnid 34379 . . . . . . . 8 (R ∈ V → ¬ {R} ∈ R)
1210, 11ax-mp 5 . . . . . . 7 ¬ {R} ∈ R
1312intnan 489 . . . . . 6 ¬ (({R‘(1st𝐴)) ∈ R ∧ {R} ∈ R)
14 opelxp 5567 . . . . . 6 (⟨({R‘(1st𝐴)), {R}⟩ ∈ (R × R) ↔ (({R‘(1st𝐴)) ∈ R ∧ {R} ∈ R))
1513, 14mtbir 325 . . . . 5 ¬ ⟨({R‘(1st𝐴)), {R}⟩ ∈ (R × R)
16 df-c 10521 . . . . . 6 ℂ = (R × R)
1716eleq2i 2902 . . . . 5 (⟨({R‘(1st𝐴)), {R}⟩ ∈ ℂ ↔ ⟨({R‘(1st𝐴)), {R}⟩ ∈ (R × R))
1815, 17mtbir 325 . . . 4 ¬ ⟨({R‘(1st𝐴)), {R}⟩ ∈ ℂ
19 eqcom 2827 . . . . . 6 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ ↔ ⟨({R‘(1st𝐴)), {R}⟩ = (+∞e𝐴))
2019biimpi 218 . . . . 5 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ → ⟨({R‘(1st𝐴)), {R}⟩ = (+∞e𝐴))
2120eleq1d 2895 . . . 4 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ → (⟨({R‘(1st𝐴)), {R}⟩ ∈ ℂ ↔ (+∞e𝐴) ∈ ℂ))
2218, 21mtbii 328 . . 3 ((+∞e𝐴) = ⟨({R‘(1st𝐴)), {R}⟩ → ¬ (+∞e𝐴) ∈ ℂ)
239, 22syl 17 . 2 (𝐴 ∈ dom +∞e → ¬ (+∞e𝐴) ∈ ℂ)
24 0ncn 10533 . . 3 ¬ ∅ ∈ ℂ
25 ndmfv 6676 . . . 4 𝐴 ∈ dom +∞e → (+∞e𝐴) = ∅)
2625eleq1d 2895 . . 3 𝐴 ∈ dom +∞e → ((+∞e𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
2724, 26mtbiri 329 . 2 𝐴 ∈ dom +∞e → ¬ (+∞e𝐴) ∈ ℂ)
2823, 27pm2.61i 184 1 ¬ (+∞e𝐴) ∈ ℂ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3473  ∅c0 4269  {csn 4543  ⟨cop 4549   × cxp 5529  dom cdm 5531  ‘cfv 6331  1st c1st 7665  Rcnr 10265  ℂcc 10513  ℝcr 10514  {Rcfractemp 34495  +∞eiτcinftyexpitau 34497 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-reg 9034  ax-inf2 9082 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-omul 8085  df-er 8267  df-ec 8269  df-qs 8273  df-ni 10272  df-pli 10273  df-mi 10274  df-lti 10275  df-plpq 10308  df-mpq 10309  df-ltpq 10310  df-enq 10311  df-nq 10312  df-erq 10313  df-plq 10314  df-mq 10315  df-1nq 10316  df-rq 10317  df-ltnq 10318  df-np 10381  df-plp 10383  df-ltp 10385  df-enr 10455  df-nr 10456  df-c 10521  df-bj-inftyexpitau 34498 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator