Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climbddf | Structured version Visualization version GIF version |
Description: A converging sequence of complex numbers is bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
climbddf.1 | ⊢ Ⅎ𝑘𝐹 |
climbddf.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
climbddf | ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1128 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → 𝑀 ∈ ℤ) | |
2 | simp2 1129 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → 𝐹 ∈ dom ⇝ ) | |
3 | nfv 1906 | . . . . . 6 ⊢ Ⅎ𝑗(𝐹‘𝑘) ∈ ℂ | |
4 | climbddf.1 | . . . . . . . 8 ⊢ Ⅎ𝑘𝐹 | |
5 | nfcv 2974 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑗 | |
6 | 4, 5 | nffv 6673 | . . . . . . 7 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
7 | nfcv 2974 | . . . . . . 7 ⊢ Ⅎ𝑘ℂ | |
8 | 6, 7 | nfel 2989 | . . . . . 6 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ |
9 | fveq2 6663 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
10 | 9 | eleq1d 2894 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ)) |
11 | 3, 8, 10 | cbvralw 3439 | . . . . 5 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ∈ ℂ) |
12 | 11 | biimpi 217 | . . . 4 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ∈ ℂ) |
13 | 12 | 3ad2ant3 1127 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ∈ ℂ) |
14 | climbddf.2 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
15 | 14 | climbdd 15016 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥) |
16 | 1, 2, 13, 15 | syl3anc 1363 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥) |
17 | nfcv 2974 | . . . . . 6 ⊢ Ⅎ𝑘abs | |
18 | 17, 6 | nffv 6673 | . . . . 5 ⊢ Ⅎ𝑘(abs‘(𝐹‘𝑗)) |
19 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑘 ≤ | |
20 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑘𝑥 | |
21 | 18, 19, 20 | nfbr 5104 | . . . 4 ⊢ Ⅎ𝑘(abs‘(𝐹‘𝑗)) ≤ 𝑥 |
22 | nfv 1906 | . . . 4 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑘)) ≤ 𝑥 | |
23 | 2fveq3 6668 | . . . . 5 ⊢ (𝑗 = 𝑘 → (abs‘(𝐹‘𝑗)) = (abs‘(𝐹‘𝑘))) | |
24 | 23 | breq1d 5067 | . . . 4 ⊢ (𝑗 = 𝑘 → ((abs‘(𝐹‘𝑗)) ≤ 𝑥 ↔ (abs‘(𝐹‘𝑘)) ≤ 𝑥)) |
25 | 21, 22, 24 | cbvralw 3439 | . . 3 ⊢ (∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
26 | 25 | rexbii 3244 | . 2 ⊢ (∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
27 | 16, 26 | sylib 219 | 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 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Ⅎwnfc 2958 ∀wral 3135 ∃wrex 3136 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 ℂcc 10523 ℝcr 10524 ≤ cle 10664 ℤcz 11969 ℤ≥cuz 12231 abscabs 14581 ⇝ cli 14829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 |
This theorem is referenced by: climinf2mpt 41871 climinf3 41873 |
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