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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-inftyexpitaudisj | Structured version Visualization version GIF version |
Description: An element of the circle at infinity is not a complex number. (Contributed by BJ, 4-Feb-2023.) |
Ref | Expression |
---|---|
bj-inftyexpitaudisj | ⊢ ¬ (+∞eiτ‘𝐴) ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2fveq3 6844 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ({R‘(1st ‘𝑥)) = ({R‘(1st ‘𝐴))) | |
2 | 1 | opeq1d 4834 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈({R‘(1st ‘𝑥)), {R}〉 = 〈({R‘(1st ‘𝐴)), {R}〉) |
3 | df-bj-inftyexpitau 35601 | . . . . 5 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ 〈({R‘(1st ‘𝑥)), {R}〉) | |
4 | opex 5419 | . . . . 5 ⊢ 〈({R‘(1st ‘𝐴)), {R}〉 ∈ V | |
5 | 2, 3, 4 | fvmpt 6945 | . . . 4 ⊢ (𝐴 ∈ ℝ → (+∞eiτ‘𝐴) = 〈({R‘(1st ‘𝐴)), {R}〉) |
6 | opex 5419 | . . . . 5 ⊢ 〈({R‘(1st ‘𝑦)), {R}〉 ∈ V | |
7 | df-bj-inftyexpitau 35601 | . . . . 5 ⊢ +∞eiτ = (𝑦 ∈ ℝ ↦ 〈({R‘(1st ‘𝑦)), {R}〉) | |
8 | 6, 7 | dmmpti 6642 | . . . 4 ⊢ dom +∞eiτ = ℝ |
9 | 5, 8 | eleq2s 2856 | . . 3 ⊢ (𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = 〈({R‘(1st ‘𝐴)), {R}〉) |
10 | nrex1 10958 | . . . . . . . 8 ⊢ R ∈ V | |
11 | bj-nsnid 35472 | . . . . . . . 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 5667 | . . . . . 6 ⊢ (〈({R‘(1st ‘𝐴)), {R}〉 ∈ (R × R) ↔ (({R‘(1st ‘𝐴)) ∈ R ∧ {R} ∈ R)) | |
15 | 13, 14 | mtbir 322 | . . . . 5 ⊢ ¬ 〈({R‘(1st ‘𝐴)), {R}〉 ∈ (R × R) |
16 | df-c 11015 | . . . . . 6 ⊢ ℂ = (R × R) | |
17 | 16 | eleq2i 2829 | . . . . 5 ⊢ (〈({R‘(1st ‘𝐴)), {R}〉 ∈ ℂ ↔ 〈({R‘(1st ‘𝐴)), {R}〉 ∈ (R × R)) |
18 | 15, 17 | mtbir 322 | . . . 4 ⊢ ¬ 〈({R‘(1st ‘𝐴)), {R}〉 ∈ ℂ |
19 | eqcom 2744 | . . . . . 6 ⊢ ((+∞eiτ‘𝐴) = 〈({R‘(1st ‘𝐴)), {R}〉 ↔ 〈({R‘(1st ‘𝐴)), {R}〉 = (+∞eiτ‘𝐴)) | |
20 | 19 | biimpi 215 | . . . . 5 ⊢ ((+∞eiτ‘𝐴) = 〈({R‘(1st ‘𝐴)), {R}〉 → 〈({R‘(1st ‘𝐴)), {R}〉 = (+∞eiτ‘𝐴)) |
21 | 20 | eleq1d 2822 | . . . 4 ⊢ ((+∞eiτ‘𝐴) = 〈({R‘(1st ‘𝐴)), {R}〉 → (〈({R‘(1st ‘𝐴)), {R}〉 ∈ ℂ ↔ (+∞eiτ‘𝐴) ∈ ℂ)) |
22 | 18, 21 | mtbii 325 | . . 3 ⊢ ((+∞eiτ‘𝐴) = 〈({R‘(1st ‘𝐴)), {R}〉 → ¬ (+∞eiτ‘𝐴) ∈ ℂ) |
23 | 9, 22 | syl 17 | . 2 ⊢ (𝐴 ∈ dom +∞eiτ → ¬ (+∞eiτ‘𝐴) ∈ ℂ) |
24 | 0ncn 11027 | . . 3 ⊢ ¬ ∅ ∈ ℂ | |
25 | ndmfv 6874 | . . . 4 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ∅) | |
26 | 25 | eleq1d 2822 | . . 3 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → ((+∞eiτ‘𝐴) ∈ ℂ ↔ ∅ ∈ ℂ)) |
27 | 24, 26 | mtbiri 326 | . 2 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → ¬ (+∞eiτ‘𝐴) ∈ ℂ) |
28 | 23, 27 | pm2.61i 182 | 1 ⊢ ¬ (+∞eiτ‘𝐴) ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∅c0 4280 {csn 4584 〈cop 4590 × cxp 5629 dom cdm 5631 ‘cfv 6493 1st c1st 7911 Rcnr 10759 ℂcc 11007 ℝcr 11008 {Rcfractemp 35598 +∞eiτcinftyexpitau 35600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-reg 9486 ax-inf2 9535 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-oadd 8408 df-omul 8409 df-er 8606 df-ec 8608 df-qs 8612 df-ni 10766 df-pli 10767 df-mi 10768 df-lti 10769 df-plpq 10802 df-mpq 10803 df-ltpq 10804 df-enq 10805 df-nq 10806 df-erq 10807 df-plq 10808 df-mq 10809 df-1nq 10810 df-rq 10811 df-ltnq 10812 df-np 10875 df-plp 10877 df-ltp 10879 df-enr 10949 df-nr 10950 df-c 11015 df-bj-inftyexpitau 35601 |
This theorem is referenced by: (None) |
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