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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 6650 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ({R‘(1st ‘𝑥)) = ({R‘(1st ‘𝐴))) | |
2 | 1 | opeq1d 4771 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈({R‘(1st ‘𝑥)), {R}〉 = 〈({R‘(1st ‘𝐴)), {R}〉) |
3 | df-bj-inftyexpitau 34614 | . . . . 5 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ 〈({R‘(1st ‘𝑥)), {R}〉) | |
4 | opex 5321 | . . . . 5 ⊢ 〈({R‘(1st ‘𝐴)), {R}〉 ∈ V | |
5 | 2, 3, 4 | fvmpt 6745 | . . . 4 ⊢ (𝐴 ∈ ℝ → (+∞eiτ‘𝐴) = 〈({R‘(1st ‘𝐴)), {R}〉) |
6 | opex 5321 | . . . . 5 ⊢ 〈({R‘(1st ‘𝑦)), {R}〉 ∈ V | |
7 | df-bj-inftyexpitau 34614 | . . . . 5 ⊢ +∞eiτ = (𝑦 ∈ ℝ ↦ 〈({R‘(1st ‘𝑦)), {R}〉) | |
8 | 6, 7 | dmmpti 6464 | . . . 4 ⊢ dom +∞eiτ = ℝ |
9 | 5, 8 | eleq2s 2908 | . . 3 ⊢ (𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = 〈({R‘(1st ‘𝐴)), {R}〉) |
10 | nrex1 10475 | . . . . . . . 8 ⊢ R ∈ V | |
11 | bj-nsnid 34486 | . . . . . . . 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 5555 | . . . . . 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 10532 | . . . . . 6 ⊢ ℂ = (R × R) | |
17 | 16 | eleq2i 2881 | . . . . 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 2805 | . . . . . 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 2874 | . . . 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 10544 | . . 3 ⊢ ¬ ∅ ∈ ℂ | |
25 | ndmfv 6675 | . . . 4 ⊢ (¬ 𝐴 ∈ dom +∞eiτ → (+∞eiτ‘𝐴) = ∅) | |
26 | 25 | eleq1d 2874 | . . 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 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 {csn 4525 〈cop 4531 × cxp 5517 dom cdm 5519 ‘cfv 6324 1st c1st 7669 Rcnr 10276 ℂcc 10524 ℝcr 10525 {Rcfractemp 34611 +∞eiτcinftyexpitau 34613 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-omul 8090 df-er 8272 df-ec 8274 df-qs 8278 df-ni 10283 df-pli 10284 df-mi 10285 df-lti 10286 df-plpq 10319 df-mpq 10320 df-ltpq 10321 df-enq 10322 df-nq 10323 df-erq 10324 df-plq 10325 df-mq 10326 df-1nq 10327 df-rq 10328 df-ltnq 10329 df-np 10392 df-plp 10394 df-ltp 10396 df-enr 10466 df-nr 10467 df-c 10532 df-bj-inftyexpitau 34614 |
This theorem is referenced by: (None) |
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