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Mirrors > Home > MPE Home > Th. List > climbdd | Structured version Visualization version GIF version |
Description: A converging sequence of complex numbers is bounded. (Contributed by Mario Carneiro, 9-Jul-2017.) |
Ref | Expression |
---|---|
climcau.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
climbdd | ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1056 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) | |
2 | climcau.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | 2 | climcau 14249 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑦 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑦) |
4 | 3 | 3adant3 1074 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∀𝑦 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑦) |
5 | 2 | caubnd 13946 | . . 3 ⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ∧ ∀𝑦 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑦) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑥) |
6 | 1, 4, 5 | syl2anc 691 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑥) |
7 | r19.26 3046 | . . . . . . 7 ⊢ (∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘(𝐹‘𝑘)) < 𝑥) ↔ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑥)) | |
8 | simpr 476 | . . . . . . . . . . 11 ⊢ (((((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ∈ ℂ) → (𝐹‘𝑘) ∈ ℂ) | |
9 | 8 | abscld 14023 | . . . . . . . . . 10 ⊢ (((((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ∈ ℂ) → (abs‘(𝐹‘𝑘)) ∈ ℝ) |
10 | simpllr 795 | . . . . . . . . . 10 ⊢ (((((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ∈ ℂ) → 𝑥 ∈ ℝ) | |
11 | ltle 10005 | . . . . . . . . . 10 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((abs‘(𝐹‘𝑘)) < 𝑥 → (abs‘(𝐹‘𝑘)) ≤ 𝑥)) | |
12 | 9, 10, 11 | syl2anc 691 | . . . . . . . . 9 ⊢ (((((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ∈ ℂ) → ((abs‘(𝐹‘𝑘)) < 𝑥 → (abs‘(𝐹‘𝑘)) ≤ 𝑥)) |
13 | 12 | expimpd 627 | . . . . . . . 8 ⊢ ((((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘(𝐹‘𝑘)) < 𝑥) → (abs‘(𝐹‘𝑘)) ≤ 𝑥)) |
14 | 13 | ralimdva 2945 | . . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ 𝑍 ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘(𝐹‘𝑘)) < 𝑥) → ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥)) |
15 | 7, 14 | syl5bir 232 | . . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑥 ∈ ℝ) → ((∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑥) → ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥)) |
16 | 15 | exp4b 630 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → (𝑥 ∈ ℝ → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑥 → ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥)))) |
17 | 16 | com23 84 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (𝑥 ∈ ℝ → (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑥 → ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥)))) |
18 | 17 | 3impia 1253 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → (𝑥 ∈ ℝ → (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑥 → ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥))) |
19 | 18 | reximdvai 2998 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑥 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥)) |
20 | 6, 19 | mpd 15 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 < clt 9953 ≤ cle 9954 − 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 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
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-rmo 2904 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 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-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 |
This theorem is referenced by: mtestbdd 23963 sge0isum 39320 |
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