Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0cnv | Structured version Visualization version GIF version |
Description: If (/) is a complex number, then it converges to itself. (see 0ncn 10543 and its comment ; see also the comment in climlimsupcex 41926) (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
0cnv | ⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (∅ ∈ ℂ → ∅ ∈ ℂ) | |
2 | 0zd 11981 | . . . . 5 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → 0 ∈ ℤ) | |
3 | simpl 483 | . . . . . . 7 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∅ ∈ ℂ) | |
4 | subid 10893 | . . . . . . . . . . 11 ⊢ (∅ ∈ ℂ → (∅ − ∅) = 0) | |
5 | 4 | fveq2d 6667 | . . . . . . . . . 10 ⊢ (∅ ∈ ℂ → (abs‘(∅ − ∅)) = (abs‘0)) |
6 | abs0 14633 | . . . . . . . . . . 11 ⊢ (abs‘0) = 0 | |
7 | 6 | a1i 11 | . . . . . . . . . 10 ⊢ (∅ ∈ ℂ → (abs‘0) = 0) |
8 | 5, 7 | eqtrd 2853 | . . . . . . . . 9 ⊢ (∅ ∈ ℂ → (abs‘(∅ − ∅)) = 0) |
9 | 8 | adantr 481 | . . . . . . . 8 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → (abs‘(∅ − ∅)) = 0) |
10 | rpgt0 12389 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → 0 < 𝑥) | |
11 | 10 | adantl 482 | . . . . . . . 8 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → 0 < 𝑥) |
12 | 9, 11 | eqbrtrd 5079 | . . . . . . 7 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → (abs‘(∅ − ∅)) < 𝑥) |
13 | 3, 12 | jca 512 | . . . . . 6 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → (∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
14 | 13 | ralrimivw 3180 | . . . . 5 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∀𝑘 ∈ (ℤ≥‘0)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
15 | fveq2 6663 | . . . . . . 7 ⊢ (𝑚 = 0 → (ℤ≥‘𝑚) = (ℤ≥‘0)) | |
16 | 15 | raleqdv 3413 | . . . . . 6 ⊢ (𝑚 = 0 → (∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘0)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥))) |
17 | 16 | rspcev 3620 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘0)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) → ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
18 | 2, 14, 17 | syl2anc 584 | . . . 4 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
19 | 18 | ralrimiva 3179 | . . 3 ⊢ (∅ ∈ ℂ → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
20 | 1, 19 | jca 512 | . 2 ⊢ (∅ ∈ ℂ → (∅ ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥))) |
21 | 0ex 5202 | . . . . 5 ⊢ ∅ ∈ V | |
22 | 21 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
23 | 0fv 6702 | . . . . 5 ⊢ (∅‘𝑘) = ∅ | |
24 | 23 | a1i 11 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℤ) → (∅‘𝑘) = ∅) |
25 | 22, 24 | clim 14839 | . . 3 ⊢ (⊤ → (∅ ⇝ ∅ ↔ (∅ ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)))) |
26 | 25 | mptru 1535 | . 2 ⊢ (∅ ⇝ ∅ ↔ (∅ ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥))) |
27 | 20, 26 | sylibr 235 | 1 ⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 Vcvv 3492 ∅c0 4288 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 < clt 10663 − cmin 10858 ℤcz 11969 ℤ≥cuz 12231 ℝ+crp 12377 abscabs 14581 ⇝ cli 14829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 |
This theorem is referenced by: climlimsupcex 41926 |
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