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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 11162 and its comment; see also the comment in climlimsupcex 45159. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
0cnv | ⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (∅ ∈ ℂ → ∅ ∈ ℂ) | |
2 | 0zd 12606 | . . . . 5 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → 0 ∈ ℤ) | |
3 | simpl 481 | . . . . . . 7 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∅ ∈ ℂ) | |
4 | subid 11515 | . . . . . . . . . . 11 ⊢ (∅ ∈ ℂ → (∅ − ∅) = 0) | |
5 | 4 | fveq2d 6904 | . . . . . . . . . 10 ⊢ (∅ ∈ ℂ → (abs‘(∅ − ∅)) = (abs‘0)) |
6 | abs0 15270 | . . . . . . . . . . 11 ⊢ (abs‘0) = 0 | |
7 | 6 | a1i 11 | . . . . . . . . . 10 ⊢ (∅ ∈ ℂ → (abs‘0) = 0) |
8 | 5, 7 | eqtrd 2767 | . . . . . . . . 9 ⊢ (∅ ∈ ℂ → (abs‘(∅ − ∅)) = 0) |
9 | 8 | adantr 479 | . . . . . . . 8 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → (abs‘(∅ − ∅)) = 0) |
10 | rpgt0 13024 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → 0 < 𝑥) | |
11 | 10 | adantl 480 | . . . . . . . 8 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → 0 < 𝑥) |
12 | 9, 11 | eqbrtrd 5172 | . . . . . . 7 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → (abs‘(∅ − ∅)) < 𝑥) |
13 | 3, 12 | jca 510 | . . . . . 6 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → (∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
14 | 13 | ralrimivw 3146 | . . . . 5 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∀𝑘 ∈ (ℤ≥‘0)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
15 | fveq2 6900 | . . . . . . 7 ⊢ (𝑚 = 0 → (ℤ≥‘𝑚) = (ℤ≥‘0)) | |
16 | 15 | raleqdv 3321 | . . . . . 6 ⊢ (𝑚 = 0 → (∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘0)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥))) |
17 | 16 | rspcev 3609 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ ∀𝑘 ∈ (ℤ≥‘0)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) → ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
18 | 2, 14, 17 | syl2anc 582 | . . . 4 ⊢ ((∅ ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
19 | 18 | ralrimiva 3142 | . . 3 ⊢ (∅ ∈ ℂ → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥)) |
20 | 1, 19 | jca 510 | . 2 ⊢ (∅ ∈ ℂ → (∅ ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑚)(∅ ∈ ℂ ∧ (abs‘(∅ − ∅)) < 𝑥))) |
21 | 0ex 5309 | . . . . 5 ⊢ ∅ ∈ V | |
22 | 21 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
23 | 0fv 6944 | . . . . 5 ⊢ (∅‘𝑘) = ∅ | |
24 | 23 | a1i 11 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℤ) → (∅‘𝑘) = ∅) |
25 | 22, 24 | clim 15476 | . . 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 394 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ∀wral 3057 ∃wrex 3066 Vcvv 3471 ∅c0 4324 class class class wbr 5150 ‘cfv 6551 (class class class)co 7424 ℂcc 11142 0cc0 11144 < clt 11284 − cmin 11480 ℤcz 12594 ℤ≥cuz 12858 ℝ+crp 13012 abscabs 15219 ⇝ cli 15466 |
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 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-seq 14005 df-exp 14065 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-clim 15470 |
This theorem is referenced by: climlimsupcex 45159 |
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