Theorem bj-inftyexpitaudisj 38220

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 6891	. . . . . 6 ⊢ (𝑥 = 𝐴 → ({R‘(1st ‘𝑥)) = ({R‘(1st ‘𝐴)))
2	1	opeq1d 4845	. . . . 5 ⊢ (𝑥 = 𝐴 → ⟨({R‘(1st ‘𝑥)), {R}⟩ = ⟨({R‘(1st ‘𝐴)), {R}⟩)
3		df-bj-inftyexpitau 38214	. . . . 5 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)
4		opex 5431	. . . . 5 ⊢ ⟨({R‘(1st ‘𝐴)), {R}⟩ ∈ V
5	2, 3, 4	fvmpt 6994	. . . 4 ⊢ (𝐴 ∈ ℝ → (+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩)
6		opex 5431	. . . . 5 ⊢ ⟨({R‘(1st ‘𝑦)), {R}⟩ ∈ V
7		df-bj-inftyexpitau 38214	. . . . 5 ⊢ +∞eiτ = (𝑦 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑦)), {R}⟩)
8	6, 7	dmmpti 6684	. . . 4 ⊢ dom +∞eiτ = ℝ
9	5, 8	eleq2s 2878	. . 3 ⊢ (𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ⟨({R‘(1st ‘𝐴)), {R}⟩)
10		nrex1 11153	. . . . . . . 8 ⊢ R ∈ V
11		bj-nsnid 38078	. . . . . . . 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 5686	. . . . . 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 11210	. . . . . 6 ⊢ ℂ = (R × R)
17	16	eleq2i 2852	. . . . 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 2767	. . . . . 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 2845	. . . 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 11222	. . 3 ⊢ ¬ ∅ ∈ ℂ
25		ndmfv 6918	. . . 4 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ∅)
26	25	eleq1d 2845	. . 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 1569   ∈ wcel 2144  Vcvv 3450  ∅c0 4278  {csn 4584  ⟨cop 4834   × cxp 5648  dom cdm 5650  ‘cfv 6539  1^st c1st 8000  Rcnr 10954  ℂcc 11202  ℝcr 11203  {_Rcfractemp 38211  +∞e^iτcinftyexpitau 38213

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-reg 9590  ax-inf2 9646

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-oadd 8484  df-omul 8485  df-er 8721  df-ec 8723  df-qs 8727  df-ni 10961  df-pli 10962  df-mi 10963  df-lti 10964  df-plpq 10997  df-mpq 10998  df-ltpq 10999  df-enq 11000  df-nq 11001  df-erq 11002  df-plq 11003  df-mq 11004  df-1nq 11005  df-rq 11006  df-ltnq 11007  df-np 11070  df-plp 11072  df-ltp 11074  df-enr 11144  df-nr 11145  df-c 11210  df-bj-inftyexpitau 38214

This theorem is used by: (None)