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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 6910 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ({R‘(1st ‘𝑥)) = ({R‘(1st ‘𝐴))) | |
| 2 | 1 | opeq1d 4878 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈({R‘(1st ‘𝑥)), {R}〉 = 〈({R‘(1st ‘𝐴)), {R}〉) | 
| 3 | df-bj-inftyexpitau 37201 | . . . . 5 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ 〈({R‘(1st ‘𝑥)), {R}〉) | |
| 4 | opex 5468 | . . . . 5 ⊢ 〈({R‘(1st ‘𝐴)), {R}〉 ∈ V | |
| 5 | 2, 3, 4 | fvmpt 7015 | . . . 4 ⊢ (𝐴 ∈ ℝ → (+∞eiτ‘𝐴) = 〈({R‘(1st ‘𝐴)), {R}〉) | 
| 6 | opex 5468 | . . . . 5 ⊢ 〈({R‘(1st ‘𝑦)), {R}〉 ∈ V | |
| 7 | df-bj-inftyexpitau 37201 | . . . . 5 ⊢ +∞eiτ = (𝑦 ∈ ℝ ↦ 〈({R‘(1st ‘𝑦)), {R}〉) | |
| 8 | 6, 7 | dmmpti 6711 | . . . 4 ⊢ dom +∞eiτ = ℝ | 
| 9 | 5, 8 | eleq2s 2858 | . . 3 ⊢ (𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = 〈({R‘(1st ‘𝐴)), {R}〉) | 
| 10 | nrex1 11105 | . . . . . . . 8 ⊢ R ∈ V | |
| 11 | bj-nsnid 37072 | . . . . . . . 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 5720 | . . . . . 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 11162 | . . . . . 6 ⊢ ℂ = (R × R) | |
| 17 | 16 | eleq2i 2832 | . . . . 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 2743 | . . . . . 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 2825 | . . . 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 11174 | . . 3 ⊢ ¬ ∅ ∈ ℂ | |
| 25 | ndmfv 6940 | . . . 4 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ∅) | |
| 26 | 25 | eleq1d 2825 | . . 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 1539 ∈ wcel 2107 Vcvv 3479 ∅c0 4332 {csn 4625 〈cop 4631 × cxp 5682 dom cdm 5684 ‘cfv 6560 1st c1st 8013 Rcnr 10906 ℂcc 11154 ℝcr 11155 {Rcfractemp 37198 +∞eiτcinftyexpitau 37200 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-reg 9633 ax-inf2 9682 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-omul 8512 df-er 8746 df-ec 8748 df-qs 8752 df-ni 10913 df-pli 10914 df-mi 10915 df-lti 10916 df-plpq 10949 df-mpq 10950 df-ltpq 10951 df-enq 10952 df-nq 10953 df-erq 10954 df-plq 10955 df-mq 10956 df-1nq 10957 df-rq 10958 df-ltnq 10959 df-np 11022 df-plp 11024 df-ltp 11026 df-enr 11096 df-nr 11097 df-c 11162 df-bj-inftyexpitau 37201 | 
| This theorem is referenced by: (None) | 
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