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Mirrors > Home > ILE Home > Th. List > climi2 | 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 11266 | . 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝐶)) |
7 | simpr 110 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝐶) → (abs‘(𝐵 − 𝐴)) < 𝐶) | |
8 | 7 | ralimi 2540 | . . 3 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝐶) → ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 𝐴)) < 𝐶) |
9 | 8 | reximi 2574 | . 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 14 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 𝐴)) < 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 class class class wbr 4000 ‘cfv 5211 (class class class)co 5868 ℂcc 7787 < clt 7969 − cmin 8105 ℤcz 9229 ℤ≥cuz 9504 ℝ+crp 9627 abscabs 10977 ⇝ cli 11257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-inn 8896 df-n0 9153 df-z 9230 df-uz 9505 df-clim 11258 |
This theorem is referenced by: climcn1 11287 climcn2 11288 climge0 11304 climsqz 11314 climsqz2 11315 mertenslem2 11515 |
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