Theorem bj-inftyexpitaudisj 34936

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 6669	. . . . . 6 ⊢ (𝑥 = 𝐴 → ({R‘(1st ‘𝑥)) = ({R‘(1st ‘𝐴)))
2	1	opeq1d 4773	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨({R‘(1st ‘𝑥)), {R}⟩ = ⟨({R‘(1st ‘𝐴)), {R}⟩)
3		df-bj-inftyexpitau 34930	. . . . 5 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)
4		opex 5329	. . . . 5 ⊢ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ V
5	2, 3, 4	fvmpt 6765	. . . 4 ⊢ (𝐴 ∈ ℝ → (+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩)
6		opex 5329	. . . . 5 ⊢ ⟨({R‘(1st ‘𝑦)), {R}⟩ ∈ V
7		df-bj-inftyexpitau 34930	. . . . 5 ⊢ +∞eiτ = (𝑦 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑦)), {R}⟩)
8	6, 7	dmmpti 6481	. . . 4 ⊢ dom +∞eiτ = ℝ
9	5, 8	eleq2s 2871	. . 3 ⊢ (𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩)
10		nrex1 10538	. . . . . . . 8 ⊢ R ∈ V
11		bj-nsnid 34802	. . . . . . . 8 ⊢ (R ∈ V → ¬ {R} ∈ R)
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ ¬ {R} ∈ R
13	12	intnan 490	. . . . . 6 ⊢ ¬ (({R‘(1st ‘𝐴)) ∈ R ∧ {R} ∈ R)
14		opelxp 5565	. . . . . 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 10595	. . . . . 6 ⊢ ℂ = (R × R)
17	16	eleq2i 2844	. . . . 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 2766	. . . . . 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 2837	. . . 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 17	. 2 ⊢ (𝐴 ∈ dom +∞eiτ → ¬ (+∞eiτ‘𝐴) ∈ ℂ)
24		0ncn 10607	. . 3 ⊢ ¬ ∅ ∈ ℂ
25		ndmfv 6694	. . . 4 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ∅)
26	25	eleq1d 2837	. . 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 185	1 ⊢ ¬ (+∞eiτ‘𝐴) ∈ ℂ

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1539   ∈ wcel 2112  Vcvv 3410  ∅c0 4228  {csn 4526  ⟨cop 4532   × cxp 5527  dom cdm 5529  ‘cfv 6341  1^st c1st 7698  Rcnr 10339  ℂcc 10587  ℝcr 10588  {_Rcfractemp 34927  +∞e^iτcinftyexpitau 34929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466  ax-reg 9103  ax-inf2 9151

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-int 4843  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6132  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-om 7587  df-1st 7700  df-2nd 7701  df-wrecs 7964  df-recs 8025  df-rdg 8063  df-1o 8119  df-oadd 8123  df-omul 8124  df-er 8306  df-ec 8308  df-qs 8312  df-ni 10346  df-pli 10347  df-mi 10348  df-lti 10349  df-plpq 10382  df-mpq 10383  df-ltpq 10384  df-enq 10385  df-nq 10386  df-erq 10387  df-plq 10388  df-mq 10389  df-1nq 10390  df-rq 10391  df-ltnq 10392  df-np 10455  df-plp 10457  df-ltp 10459  df-enr 10529  df-nr 10530  df-c 10595  df-bj-inftyexpitau 34930

This theorem is referenced by: (None)