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 14849 | . . . . . . 7 ⊢ Rel ⇝ | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → Rel ⇝ ) |
9 | brrelex1 5605 | . . . . . 6 ⊢ ((Rel ⇝ ∧ 𝐹 ⇝ 𝐴) → 𝐹 ∈ V) | |
10 | 8, 2, 9 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ V) |
11 | clim2d.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) | |
12 | 3, 4, 5, 6, 10, 11 | clim2f2 41971 | . . . 4 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))) |
13 | 2, 12 | mpbid 234 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))) |
14 | 13 | simprd 498 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)) |
15 | breq2 5070 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((abs‘(𝐵 − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝑋)) | |
16 | 15 | anbi2d 630 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋))) |
17 | 16 | ralbidv 3197 | . . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋))) |
18 | 17 | rexbidv 3297 | . . 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 3621 | . 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 586 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 ∀wral 3138 ∃wrex 3139 Vcvv 3494 class class class wbr 5066 Rel wrel 5560 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 < clt 10675 − cmin 10870 ℤcz 11982 ℤ≥cuz 12244 ℝ+crp 12390 abscabs 14593 ⇝ cli 14841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-neg 10873 df-z 11983 df-uz 12245 df-clim 14845 |
This theorem is referenced by: climleltrp 41977 |
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