Theorem bj-inftyexpitaudisj 37882

Description: An element of the circle at infinity is not a complex number. (Contributed by BJ, 4-Feb-2023.)

Assertion

Ref	Expression
bj-inftyexpitaudisj	⊢ ¬ (+∞eiτ‘𝐴) ∈ ℂ

Proof of Theorem bj-inftyexpitaudisj

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6890	. . . . . 6 ⊢ (𝑥 = 𝐴 → ({R‘(1st ‘𝑥)) = ({R‘(1st ‘𝐴)))
2	1	opeq1d 4846	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨({R‘(1st ‘𝑥)), {R}⟩ = ⟨({R‘(1st ‘𝐴)), {R}⟩)
3		df-bj-inftyexpitau 37876	. . . . 5 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)
4		opex 5447	. . . . 5 ⊢ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ V
5	2, 3, 4	fvmpt 6993	. . . 4 ⊢ (𝐴 ∈ ℝ → (+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩)
6		opex 5447	. . . . 5 ⊢ ⟨({R‘(1st ‘𝑦)), {R}⟩ ∈ V
7		df-bj-inftyexpitau 37876	. . . . 5 ⊢ +∞eiτ = (𝑦 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑦)), {R}⟩)
8	6, 7	dmmpti 6683	. . . 4 ⊢ dom +∞eiτ = ℝ
9	5, 8	eleq2s 2883	. . 3 ⊢ (𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩)
10		nrex1 11060	. . . . . . . 8 ⊢ R ∈ V
11		bj-nsnid 37739	. . . . . . . 8 ⊢ (R ∈ V → ¬ {R} ∈ R)
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ ¬ {R} ∈ R
13	12	intnan 492	. . . . . 6 ⊢ ¬ (({R‘(1st ‘𝐴)) ∈ R ∧ {R} ∈ R)
14		opelxp 5699	. . . . . 6 ⊢ (⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ (R × R) ↔ (({R‘(1st ‘𝐴)) ∈ R ∧ {R} ∈ R))
15	13, 14	mtbir 326	. . . . 5 ⊢ ¬ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ (R × R)
16		df-c 11117	. . . . . 6 ⊢ ℂ = (R × R)
17	16	eleq2i 2857	. . . . 5 ⊢ (⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ ℂ ↔ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ (R × R))
18	15, 17	mtbir 326	. . . 4 ⊢ ¬ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ ℂ
19		eqcom 2772	. . . . . 6 ⊢ ((+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩ ↔ ⟨({R‘(1st ‘𝐴)), {R}⟩ = (+∞eiτ‘𝐴))
20	19	biimpi 219	. . . . 5 ⊢ ((+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩ → ⟨({R‘(1st ‘𝐴)), {R}⟩ = (+∞eiτ‘𝐴))
21	20	eleq1d 2850	. . . 4 ⊢ ((+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩ → (⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ ℂ ↔ (+∞eiτ‘𝐴) ∈ ℂ))
22	18, 21	mtbii 329	. . 3 ⊢ ((+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩ → ¬ (+∞eiτ‘𝐴) ∈ ℂ)
23	9, 22	syl 18	. 2 ⊢ (𝐴 ∈ dom +∞eiτ → ¬ (+∞eiτ‘𝐴) ∈ ℂ)
24		0ncn 11129	. . 3 ⊢ ¬ ∅ ∈ ℂ
25		ndmfv 6917	. . . 4 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ∅)
26	25	eleq1d 2850	. . 3 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → ((+∞eiτ‘𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
27	24, 26	mtbiri 330	. 2 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → ¬ (+∞eiτ‘𝐴) ∈ ℂ)
28	23, 27	pm2.61i 184	1 ⊢ ¬ (+∞eiτ‘𝐴) ∈ ℂ

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ∅c0 4286  {csn 4591  ⟨cop 4597   × cxp 5661  dom cdm 5663  ‘cfv 6540  1^st c1st 7986  Rcnr 10861  ℂcc 11109  ℝcr 11110  {_Rcfractemp 37873  +∞e^iτcinftyexpitau 37875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-reg 9557  ax-inf2 9613

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oadd 8459  df-omul 8460  df-er 8696  df-ec 8698  df-qs 8702  df-ni 10868  df-pli 10869  df-mi 10870  df-lti 10871  df-plpq 10904  df-mpq 10905  df-ltpq 10906  df-enq 10907  df-nq 10908  df-erq 10909  df-plq 10910  df-mq 10911  df-1nq 10912  df-rq 10913  df-ltnq 10914  df-np 10977  df-plp 10979  df-ltp 10981  df-enr 11051  df-nr 11052  df-c 11117  df-bj-inftyexpitau 37876

This theorem is used by: (None)