Theorem bj-inftyexpitaudisj 37572

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 6839	. . . . . 6 ⊢ (𝑥 = 𝐴 → ({R‘(1st ‘𝑥)) = ({R‘(1st ‘𝐴)))
2	1	opeq1d 4817	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨({R‘(1st ‘𝑥)), {R}⟩ = ⟨({R‘(1st ‘𝐴)), {R}⟩)
3		df-bj-inftyexpitau 37566	. . . . 5 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)
4		opex 5410	. . . . 5 ⊢ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ V
5	2, 3, 4	fvmpt 6942	. . . 4 ⊢ (𝐴 ∈ ℝ → (+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩)
6		opex 5410	. . . . 5 ⊢ ⟨({R‘(1st ‘𝑦)), {R}⟩ ∈ V
7		df-bj-inftyexpitau 37566	. . . . 5 ⊢ +∞eiτ = (𝑦 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑦)), {R}⟩)
8	6, 7	dmmpti 6636	. . . 4 ⊢ dom +∞eiτ = ℝ
9	5, 8	eleq2s 2858	. . 3 ⊢ (𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩)
10		nrex1 10985	. . . . . . . 8 ⊢ R ∈ V
11		bj-nsnid 37430	. . . . . . . 8 ⊢ (R ∈ V → ¬ {R} ∈ R)
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ ¬ {R} ∈ R
13	12	intnan 487	. . . . . 6 ⊢ ¬ (({R‘(1st ‘𝐴)) ∈ R ∧ {R} ∈ R)
14		opelxp 5661	. . . . . 6 ⊢ (⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ (R × R) ↔ (({R‘(1st ‘𝐴)) ∈ R ∧ {R} ∈ R))
15	13, 14	mtbir 324	. . . . 5 ⊢ ¬ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ (R × R)
16		df-c 11042	. . . . . 6 ⊢ ℂ = (R × R)
17	16	eleq2i 2832	. . . . 5 ⊢ (⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ ℂ ↔ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ (R × R))
18	15, 17	mtbir 324	. . . 4 ⊢ ¬ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ ℂ
19		eqcom 2747	. . . . . 6 ⊢ ((+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩ ↔ ⟨({R‘(1st ‘𝐴)), {R}⟩ = (+∞eiτ‘𝐴))
20	19	biimpi 217	. . . . 5 ⊢ ((+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩ → ⟨({R‘(1st ‘𝐴)), {R}⟩ = (+∞eiτ‘𝐴))
21	20	eleq1d 2825	. . . 4 ⊢ ((+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩ → (⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ ℂ ↔ (+∞eiτ‘𝐴) ∈ ℂ))
22	18, 21	mtbii 327	. . 3 ⊢ ((+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩ → ¬ (+∞eiτ‘𝐴) ∈ ℂ)
23	9, 22	syl 17	. 2 ⊢ (𝐴 ∈ dom +∞eiτ → ¬ (+∞eiτ‘𝐴) ∈ ℂ)
24		0ncn 11054	. . 3 ⊢ ¬ ∅ ∈ ℂ
25		ndmfv 6866	. . . 4 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ∅)
26	25	eleq1d 2825	. . 3 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → ((+∞eiτ‘𝐴) ∈ ℂ ↔ ∅ ∈ ℂ))
27	24, 26	mtbiri 328	. 2 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → ¬ (+∞eiτ‘𝐴) ∈ ℂ)
28	23, 27	pm2.61i 183	1 ⊢ ¬ (+∞eiτ‘𝐴) ∈ ℂ

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432  ∅c0 4268  {csn 4562  ⟨cop 4568   × cxp 5623  dom cdm 5625  ‘cfv 6492  1^st c1st 7936  Rcnr 10786  ℂcc 11034  ℝcr 11035  {_Rcfractemp 37563  +∞e^iτcinftyexpitau 37565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-reg 9504  ax-inf2 9560

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-oadd 8406  df-omul 8407  df-er 8640  df-ec 8642  df-qs 8646  df-ni 10793  df-pli 10794  df-mi 10795  df-lti 10796  df-plpq 10829  df-mpq 10830  df-ltpq 10831  df-enq 10832  df-nq 10833  df-erq 10834  df-plq 10835  df-mq 10836  df-1nq 10837  df-rq 10838  df-ltnq 10839  df-np 10902  df-plp 10904  df-ltp 10906  df-enr 10976  df-nr 10977  df-c 11042  df-bj-inftyexpitau 37566

This theorem is referenced by: (None)