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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climd | Structured version Visualization version GIF version |
Description: Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
climd.1 | ⊢ Ⅎ𝑘𝜑 |
climd.2 | ⊢ Ⅎ𝑘𝐹 |
climd.3 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climd.4 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climd.5 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climd.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) |
climd.7 | ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
Ref | Expression |
---|---|
climd | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climd.7 | . 2 ⊢ (𝜑 → 𝑋 ∈ ℝ+) | |
2 | climd.5 | . . . 4 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
3 | climd.1 | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
4 | climd.2 | . . . . 5 ⊢ Ⅎ𝑘𝐹 | |
5 | climd.3 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | climd.4 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | climrel 15440 | . . . . . . 7 ⊢ Rel ⇝ | |
8 | 7 | brrelex1i 5732 | . . . . . 6 ⊢ (𝐹 ⇝ 𝐴 → 𝐹 ∈ V) |
9 | 2, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ V) |
10 | climd.6 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) | |
11 | 3, 4, 5, 6, 9, 10 | clim2f2 44685 | . . . 4 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))) |
12 | 2, 11 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))) |
13 | 12 | simprd 496 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)) |
14 | breq2 5152 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((abs‘(𝐵 − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝑋)) | |
15 | 14 | anbi2d 629 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋))) |
16 | 15 | rexralbidv 3220 | . . 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))) |
17 | 16 | rspcva 3610 | . 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)) |
18 | 1, 13, 17 | syl2anc 584 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 ∀wral 3061 ∃wrex 3070 Vcvv 3474 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 < clt 11252 − cmin 11448 ℤcz 12562 ℤ≥cuz 12826 ℝ+crp 12978 abscabs 15185 ⇝ cli 15432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-neg 11451 df-z 12563 df-uz 12827 df-clim 15436 |
This theorem is referenced by: fnlimabslt 44694 |
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