Theorem bj-inftyexpitaudisj 37193

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 6863	. . . . . 6 ⊢ (𝑥 = 𝐴 → ({R‘(1st ‘𝑥)) = ({R‘(1st ‘𝐴)))
2	1	opeq1d 4843	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨({R‘(1st ‘𝑥)), {R}⟩ = ⟨({R‘(1st ‘𝐴)), {R}⟩)
3		df-bj-inftyexpitau 37187	. . . . 5 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)
4		opex 5424	. . . . 5 ⊢ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ V
5	2, 3, 4	fvmpt 6968	. . . 4 ⊢ (𝐴 ∈ ℝ → (+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩)
6		opex 5424	. . . . 5 ⊢ ⟨({R‘(1st ‘𝑦)), {R}⟩ ∈ V
7		df-bj-inftyexpitau 37187	. . . . 5 ⊢ +∞eiτ = (𝑦 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑦)), {R}⟩)
8	6, 7	dmmpti 6662	. . . 4 ⊢ dom +∞eiτ = ℝ
9	5, 8	eleq2s 2846	. . 3 ⊢ (𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩)
10		nrex1 11017	. . . . . . . 8 ⊢ R ∈ V
11		bj-nsnid 37058	. . . . . . . 8 ⊢ (R ∈ V → ¬ {R} ∈ R)
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ ¬ {R} ∈ R
13	12	intnan 486	. . . . . 6 ⊢ ¬ (({R‘(1st ‘𝐴)) ∈ R ∧ {R} ∈ R)
14		opelxp 5674	. . . . . 6 ⊢ (⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ (R × R) ↔ (({R‘(1st ‘𝐴)) ∈ R ∧ {R} ∈ R))
15	13, 14	mtbir 323	. . . . 5 ⊢ ¬ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ (R × R)
16		df-c 11074	. . . . . 6 ⊢ ℂ = (R × R)
17	16	eleq2i 2820	. . . . 5 ⊢ (⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ ℂ ↔ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ (R × R))
18	15, 17	mtbir 323	. . . 4 ⊢ ¬ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ ℂ
19		eqcom 2736	. . . . . 6 ⊢ ((+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩ ↔ ⟨({R‘(1st ‘𝐴)), {R}⟩ = (+∞eiτ‘𝐴))
20	19	biimpi 216	. . . . 5 ⊢ ((+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩ → ⟨({R‘(1st ‘𝐴)), {R}⟩ = (+∞eiτ‘𝐴))
21	20	eleq1d 2813	. . . 4 ⊢ ((+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩ → (⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ ℂ ↔ (+∞eiτ‘𝐴) ∈ ℂ))
22	18, 21	mtbii 326	. . 3 ⊢ ((+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩ → ¬ (+∞eiτ‘𝐴) ∈ ℂ)
23	9, 22	syl 17	. 2 ⊢ (𝐴 ∈ dom +∞eiτ → ¬ (+∞eiτ‘𝐴) ∈ ℂ)
24		0ncn 11086	. . 3 ⊢ ¬ ∅ ∈ ℂ
25		ndmfv 6893	. . . 4 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ∅)
26	25	eleq1d 2813	. . 3 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → ((+∞eiτ‘𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
27	24, 26	mtbiri 327	. 2 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → ¬ (+∞eiτ‘𝐴) ∈ ℂ)
28	23, 27	pm2.61i 182	1 ⊢ ¬ (+∞eiτ‘𝐴) ∈ ℂ

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ∅c0 4296  {csn 4589  ⟨cop 4595   × cxp 5636  dom cdm 5638  ‘cfv 6511  1^st c1st 7966  Rcnr 10818  ℂcc 11066  ℝcr 11067  {_Rcfractemp 37184  +∞e^iτcinftyexpitau 37186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-reg 9545  ax-inf2 9594

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oadd 8438  df-omul 8439  df-er 8671  df-ec 8673  df-qs 8677  df-ni 10825  df-pli 10826  df-mi 10827  df-lti 10828  df-plpq 10861  df-mpq 10862  df-ltpq 10863  df-enq 10864  df-nq 10865  df-erq 10866  df-plq 10867  df-mq 10868  df-1nq 10869  df-rq 10870  df-ltnq 10871  df-np 10934  df-plp 10936  df-ltp 10938  df-enr 11008  df-nr 11009  df-c 11074  df-bj-inftyexpitau 37187

This theorem is referenced by: (None)