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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > clim2d | Structured version Visualization version GIF version |
Description: The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
clim2d.k | ⊢ Ⅎ𝑘𝜑 |
clim2d.f | ⊢ Ⅎ𝑘𝐹 |
clim2d.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
clim2d.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
clim2d.c | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
clim2d.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) |
clim2d.x | ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
Ref | Expression |
---|---|
clim2d | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clim2d.x | . 2 ⊢ (𝜑 → 𝑋 ∈ ℝ+) | |
2 | clim2d.c | . . . 4 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
3 | clim2d.k | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
4 | clim2d.f | . . . . 5 ⊢ Ⅎ𝑘𝐹 | |
5 | clim2d.z | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | clim2d.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | climrel 14600 | . . . . . . 7 ⊢ Rel ⇝ | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → Rel ⇝ ) |
9 | brrelex1 5390 | . . . . . 6 ⊢ ((Rel ⇝ ∧ 𝐹 ⇝ 𝐴) → 𝐹 ∈ V) | |
10 | 8, 2, 9 | syl2anc 581 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ V) |
11 | clim2d.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) | |
12 | 3, 4, 5, 6, 10, 11 | clim2f2 40697 | . . . 4 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))) |
13 | 2, 12 | mpbid 224 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))) |
14 | 13 | simprd 491 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)) |
15 | breq2 4877 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((abs‘(𝐵 − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝑋)) | |
16 | 15 | anbi2d 624 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋))) |
17 | 16 | ralbidv 3195 | . . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋))) |
18 | 17 | rexbidv 3262 | . . 3 ⊢ (𝑥 = 𝑋 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋))) |
19 | 18 | rspcva 3524 | . 2 ⊢ ((𝑋 ∈ ℝ+ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋)) |
20 | 1, 14, 19 | syl2anc 581 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 Ⅎwnf 1884 ∈ wcel 2166 Ⅎwnfc 2956 ∀wral 3117 ∃wrex 3118 Vcvv 3414 class class class wbr 4873 Rel wrel 5347 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 < clt 10391 − cmin 10585 ℤcz 11704 ℤ≥cuz 11968 ℝ+crp 12112 abscabs 14351 ⇝ cli 14592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-pre-lttri 10326 ax-pre-lttrn 10327 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-neg 10588 df-z 11705 df-uz 11969 df-clim 14596 |
This theorem is referenced by: climleltrp 40703 |
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