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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 15442 | . . . . . . 7 ⊢ Rel ⇝ | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → Rel ⇝ ) |
9 | brrelex1 5730 | . . . . . 6 ⊢ ((Rel ⇝ ∧ 𝐹 ⇝ 𝐴) → 𝐹 ∈ V) | |
10 | 8, 2, 9 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ V) |
11 | clim2d.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) | |
12 | 3, 4, 5, 6, 10, 11 | clim2f2 44686 | . . . 4 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))) |
13 | 2, 12 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))) |
14 | 13 | simprd 494 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)) |
15 | breq2 5153 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((abs‘(𝐵 − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝑋)) | |
16 | 15 | anbi2d 627 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋))) |
17 | 16 | ralbidv 3175 | . . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋))) |
18 | 17 | rexbidv 3176 | . . 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 3611 | . 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 582 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 Ⅎwnf 1783 ∈ wcel 2104 Ⅎwnfc 2881 ∀wral 3059 ∃wrex 3068 Vcvv 3472 class class class wbr 5149 Rel wrel 5682 ‘cfv 6544 (class class class)co 7413 ℂcc 11112 < clt 11254 − cmin 11450 ℤcz 12564 ℤ≥cuz 12828 ℝ+crp 12980 abscabs 15187 ⇝ cli 15434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-pre-lttri 11188 ax-pre-lttrn 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7416 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-neg 11453 df-z 12565 df-uz 12829 df-clim 15438 |
This theorem is referenced by: climleltrp 44692 |
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