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Mirrors > Home > MPE Home > Th. List > climi2 | Structured version Visualization version 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 |
---|---|
climi2 | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 𝐴)) < 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 | climi.5 | . . 3 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
6 | 1, 2, 3, 4, 5 | climi 14957 | . 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝐶)) |
7 | simpr 488 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝐶) → (abs‘(𝐵 − 𝐴)) < 𝐶) | |
8 | 7 | ralimi 3075 | . . 3 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝐶) → ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 𝐴)) < 𝐶) |
9 | 8 | reximi 3157 | . 2 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝐶) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 𝐴)) < 𝐶) |
10 | 6, 9 | syl 17 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 𝐴)) < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∀wral 3053 ∃wrex 3054 class class class wbr 5030 ‘cfv 6339 (class class class)co 7170 ℂcc 10613 < clt 10753 − cmin 10948 ℤcz 12062 ℤ≥cuz 12324 ℝ+crp 12472 abscabs 14683 ⇝ cli 14931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-pre-lttri 10689 ax-pre-lttrn 10690 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-neg 10951 df-z 12063 df-uz 12325 df-clim 14935 |
This theorem is referenced by: rlimclim 14993 climcn1 15039 climcn2 15040 climsqz 15088 climsqz2 15089 mertenslem2 15333 uniioombllem6 24340 ulmcau 25142 ulmdvlem3 25149 rrncmslem 35613 cvgdvgrat 41469 stoweidlem7 43090 |
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