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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 6912 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ({R‘(1st ‘𝑥)) = ({R‘(1st ‘𝐴))) | |
2 | 1 | opeq1d 4884 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈({R‘(1st ‘𝑥)), {R}〉 = 〈({R‘(1st ‘𝐴)), {R}〉) |
3 | df-bj-inftyexpitau 37182 | . . . . 5 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ 〈({R‘(1st ‘𝑥)), {R}〉) | |
4 | opex 5475 | . . . . 5 ⊢ 〈({R‘(1st ‘𝐴)), {R}〉 ∈ V | |
5 | 2, 3, 4 | fvmpt 7016 | . . . 4 ⊢ (𝐴 ∈ ℝ → (+∞eiτ‘𝐴) = 〈({R‘(1st ‘𝐴)), {R}〉) |
6 | opex 5475 | . . . . 5 ⊢ 〈({R‘(1st ‘𝑦)), {R}〉 ∈ V | |
7 | df-bj-inftyexpitau 37182 | . . . . 5 ⊢ +∞eiτ = (𝑦 ∈ ℝ ↦ 〈({R‘(1st ‘𝑦)), {R}〉) | |
8 | 6, 7 | dmmpti 6713 | . . . 4 ⊢ dom +∞eiτ = ℝ |
9 | 5, 8 | eleq2s 2857 | . . 3 ⊢ (𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = 〈({R‘(1st ‘𝐴)), {R}〉) |
10 | nrex1 11102 | . . . . . . . 8 ⊢ R ∈ V | |
11 | bj-nsnid 37053 | . . . . . . . 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 5725 | . . . . . 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 11159 | . . . . . 6 ⊢ ℂ = (R × R) | |
17 | 16 | eleq2i 2831 | . . . . 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 2742 | . . . . . 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 2824 | . . . 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 11171 | . . 3 ⊢ ¬ ∅ ∈ ℂ | |
25 | ndmfv 6942 | . . . 4 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ∅) | |
26 | 25 | eleq1d 2824 | . . 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 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 〈cop 4637 × cxp 5687 dom cdm 5689 ‘cfv 6563 1st c1st 8011 Rcnr 10903 ℂcc 11151 ℝcr 11152 {Rcfractemp 37179 +∞eiτcinftyexpitau 37181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-reg 9630 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-omul 8510 df-er 8744 df-ec 8746 df-qs 8750 df-ni 10910 df-pli 10911 df-mi 10912 df-lti 10913 df-plpq 10946 df-mpq 10947 df-ltpq 10948 df-enq 10949 df-nq 10950 df-erq 10951 df-plq 10952 df-mq 10953 df-1nq 10954 df-rq 10955 df-ltnq 10956 df-np 11019 df-plp 11021 df-ltp 11023 df-enr 11093 df-nr 11094 df-c 11159 df-bj-inftyexpitau 37182 |
This theorem is referenced by: (None) |
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