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Mirrors > Home > ILE Home > Th. List > climi | GIF version |
Description: Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
climi.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climi.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climi.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
climi.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) |
climi.5 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
Ref | Expression |
---|---|
climi | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 3933 | . . . 4 ⊢ (𝑥 = 𝐶 → ((abs‘(𝐵 − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝐶)) | |
2 | 1 | anbi2d 459 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ (𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝐶))) |
3 | 2 | rexralbidv 2461 | . 2 ⊢ (𝑥 = 𝐶 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝐶))) |
4 | climi.5 | . . . 4 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
5 | climi.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | climi.2 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | climrel 11049 | . . . . . . 7 ⊢ Rel ⇝ | |
8 | 7 | brrelex1i 4582 | . . . . . 6 ⊢ (𝐹 ⇝ 𝐴 → 𝐹 ∈ V) |
9 | 4, 8 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ V) |
10 | climi.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) | |
11 | 5, 6, 9, 10 | clim2 11052 | . . . 4 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)))) |
12 | 4, 11 | mpbid 146 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))) |
13 | 12 | simprd 113 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥)) |
14 | climi.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
15 | 3, 13, 14 | rspcdva 2794 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 Vcvv 2686 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 < clt 7800 − cmin 7933 ℤcz 9054 ℤ≥cuz 9326 ℝ+crp 9441 abscabs 10769 ⇝ cli 11047 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-clim 11048 |
This theorem is referenced by: climi2 11057 climi0 11058 climuni 11062 2clim 11070 climcau 11116 climcaucn 11120 |
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