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 14071 | . . . . . . 7 ⊢ Rel ⇝ | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → Rel ⇝ ) |
9 | brrelex 5080 | . . . . . 6 ⊢ ((Rel ⇝ ∧ 𝐹 ⇝ 𝐴) → 𝐹 ∈ V) | |
10 | 8, 2, 9 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ V) |
11 | clim2d.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) | |
12 | 3, 4, 5, 6, 10, 11 | clim2f2 38737 | . . . 4 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))) |
13 | 2, 12 | mpbid 221 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))) |
14 | 13 | simprd 478 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)) |
15 | breq2 4587 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((abs‘(𝐵 − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝑋)) | |
16 | 15 | anbi2d 736 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋))) |
17 | 16 | ralbidv 2969 | . . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋))) |
18 | 17 | rexbidv 3034 | . . 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 3280 | . 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 691 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 ∀wral 2896 ∃wrex 2897 Vcvv 3173 class class class wbr 4583 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 < clt 9953 − cmin 10145 ℤcz 11254 ℤ≥cuz 11563 ℝ+crp 11708 abscabs 13822 ⇝ cli 14063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-neg 10148 df-z 11255 df-uz 11564 df-clim 14067 |
This theorem is referenced by: climleltrp 38743 |
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