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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 10544 and its comment; see also the comment in climlimsupcex 42411. (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 486 | . . . . . . 7 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∅ ∈ ℂ) | |
4 | subid 10894 | . . . . . . . . . . 11 ⊢ (∅ ∈ ℂ → (∅ − ∅) = 0) | |
5 | 4 | fveq2d 6649 | . . . . . . . . . 10 ⊢ (∅ ∈ ℂ → (abs‘(∅ − ∅)) = (abs‘0)) |
6 | abs0 14637 | . . . . . . . . . . 11 ⊢ (abs‘0) = 0 | |
7 | 6 | a1i 11 | . . . . . . . . . 10 ⊢ (∅ ∈ ℂ → (abs‘0) = 0) |
8 | 5, 7 | eqtrd 2833 | . . . . . . . . 9 ⊢ (∅ ∈ ℂ → (abs‘(∅ − ∅)) = 0) |
9 | 8 | adantr 484 | . . . . . . . 8 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → (abs‘(∅ − ∅)) = 0) |
10 | rpgt0 12389 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → 0 < 𝑥) | |
11 | 10 | adantl 485 | . . . . . . . 8 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → 0 < 𝑥) |
12 | 9, 11 | eqbrtrd 5052 | . . . . . . 7 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → (abs‘(∅ − ∅)) < 𝑥) |
13 | 3, 12 | jca 515 | . . . . . 6 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → (∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
14 | 13 | ralrimivw 3150 | . . . . 5 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∀𝑘 ∈ (ℤ≥‘0)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
15 | fveq2 6645 | . . . . . . 7 ⊢ (𝑚 = 0 → (ℤ≥‘𝑚) = (ℤ≥‘0)) | |
16 | 15 | raleqdv 3364 | . . . . . 6 ⊢ (𝑚 = 0 → (∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘0)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥))) |
17 | 16 | rspcev 3571 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘0)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) → ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
18 | 2, 14, 17 | syl2anc 587 | . . . 4 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
19 | 18 | ralrimiva 3149 | . . 3 ⊢ (∅ ∈ ℂ → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
20 | 1, 19 | jca 515 | . 2 ⊢ (∅ ∈ ℂ → (∅ ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥))) |
21 | 0ex 5175 | . . . . 5 ⊢ ∅ ∈ V | |
22 | 21 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
23 | 0fv 6684 | . . . . 5 ⊢ (∅‘𝑘) = ∅ | |
24 | 23 | a1i 11 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℤ) → (∅‘𝑘) = ∅) |
25 | 22, 24 | clim 14843 | . . 3 ⊢ (⊤ → (∅ ⇝ ∅ ↔ (∅ ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)))) |
26 | 25 | mptru 1545 | . 2 ⊢ (∅ ⇝ ∅ ↔ (∅ ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥))) |
27 | 20, 26 | sylibr 237 | 1 ⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ∅c0 4243 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 < clt 10664 − cmin 10859 ℤcz 11969 ℤ≥cuz 12231 ℝ+crp 12377 abscabs 14585 ⇝ cli 14833 |
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-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
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-nel 3092 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-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-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 |
This theorem is referenced by: climlimsupcex 42411 |
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