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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 11125 and its comment; see also the comment in climlimsupcex 44995. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
0cnv | ⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (∅ ∈ ℂ → ∅ ∈ ℂ) | |
2 | 0zd 12568 | . . . . 5 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → 0 ∈ ℤ) | |
3 | simpl 482 | . . . . . . 7 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∅ ∈ ℂ) | |
4 | subid 11477 | . . . . . . . . . . 11 ⊢ (∅ ∈ ℂ → (∅ − ∅) = 0) | |
5 | 4 | fveq2d 6886 | . . . . . . . . . 10 ⊢ (∅ ∈ ℂ → (abs‘(∅ − ∅)) = (abs‘0)) |
6 | abs0 15230 | . . . . . . . . . . 11 ⊢ (abs‘0) = 0 | |
7 | 6 | a1i 11 | . . . . . . . . . 10 ⊢ (∅ ∈ ℂ → (abs‘0) = 0) |
8 | 5, 7 | eqtrd 2764 | . . . . . . . . 9 ⊢ (∅ ∈ ℂ → (abs‘(∅ − ∅)) = 0) |
9 | 8 | adantr 480 | . . . . . . . 8 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → (abs‘(∅ − ∅)) = 0) |
10 | rpgt0 12984 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → 0 < 𝑥) | |
11 | 10 | adantl 481 | . . . . . . . 8 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → 0 < 𝑥) |
12 | 9, 11 | eqbrtrd 5161 | . . . . . . 7 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → (abs‘(∅ − ∅)) < 𝑥) |
13 | 3, 12 | jca 511 | . . . . . 6 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → (∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
14 | 13 | ralrimivw 3142 | . . . . 5 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∀𝑘 ∈ (ℤ≥‘0)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
15 | fveq2 6882 | . . . . . . 7 ⊢ (𝑚 = 0 → (ℤ≥‘𝑚) = (ℤ≥‘0)) | |
16 | 15 | raleqdv 3317 | . . . . . 6 ⊢ (𝑚 = 0 → (∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘0)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥))) |
17 | 16 | rspcev 3604 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘0)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) → ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
18 | 2, 14, 17 | syl2anc 583 | . . . 4 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
19 | 18 | ralrimiva 3138 | . . 3 ⊢ (∅ ∈ ℂ → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
20 | 1, 19 | jca 511 | . 2 ⊢ (∅ ∈ ℂ → (∅ ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥))) |
21 | 0ex 5298 | . . . . 5 ⊢ ∅ ∈ V | |
22 | 21 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
23 | 0fv 6926 | . . . . 5 ⊢ (∅‘𝑘) = ∅ | |
24 | 23 | a1i 11 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℤ) → (∅‘𝑘) = ∅) |
25 | 22, 24 | clim 15436 | . . 3 ⊢ (⊤ → (∅ ⇝ ∅ ↔ (∅ ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)))) |
26 | 25 | mptru 1540 | . 2 ⊢ (∅ ⇝ ∅ ↔ (∅ ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥))) |
27 | 20, 26 | sylibr 233 | 1 ⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 Vcvv 3466 ∅c0 4315 class class class wbr 5139 ‘cfv 6534 (class class class)co 7402 ℂcc 11105 0cc0 11107 < clt 11246 − cmin 11442 ℤcz 12556 ℤ≥cuz 12820 ℝ+crp 12972 abscabs 15179 ⇝ cli 15426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-n0 12471 df-z 12557 df-uz 12821 df-rp 12973 df-seq 13965 df-exp 14026 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-clim 15430 |
This theorem is referenced by: climlimsupcex 44995 |
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