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Mirrors > Home > MPE Home > Th. List > climi0 | Structured version Visualization version GIF version |
Description: Convergence of a sequence of complex numbers to zero. (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) |
climi0.5 | ⊢ (𝜑 → 𝐹 ⇝ 0) |
Ref | Expression |
---|---|
climi0 | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘𝐵) < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climi.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climi.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climi.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
4 | climi.4 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) | |
5 | climi0.5 | . . 3 ⊢ (𝜑 → 𝐹 ⇝ 0) | |
6 | 1, 2, 3, 4, 5 | climi 14089 | . 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 0)) < 𝐶)) |
7 | subid1 10180 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (𝐵 − 0) = 𝐵) | |
8 | 7 | fveq2d 6107 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (abs‘(𝐵 − 0)) = (abs‘𝐵)) |
9 | 8 | breq1d 4593 | . . . . 5 ⊢ (𝐵 ∈ ℂ → ((abs‘(𝐵 − 0)) < 𝐶 ↔ (abs‘𝐵) < 𝐶)) |
10 | 9 | biimpa 500 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 0)) < 𝐶) → (abs‘𝐵) < 𝐶) |
11 | 10 | ralimi 2936 | . . 3 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 0)) < 𝐶) → ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘𝐵) < 𝐶) |
12 | 11 | reximi 2994 | . 2 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 0)) < 𝐶) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘𝐵) < 𝐶) |
13 | 6, 12 | syl 17 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘𝐵) < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 < 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-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
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-reu 2903 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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 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-sub 10147 df-neg 10148 df-z 11255 df-uz 11564 df-clim 14067 |
This theorem is referenced by: mertenslem2 14456 iscmet3lem3 22896 radcnvlem1 23971 abelthlem5 23993 abelthlem8 23997 sinccvg 30821 |
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