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 1061 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → 𝑀 ∈ ℤ) | |
2 | simp2 1062 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → 𝐹 ∈ dom ⇝ ) | |
3 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑗(𝐹‘𝑘) ∈ ℂ | |
4 | climbddf.1 | . . . . . . . 8 ⊢ Ⅎ𝑘𝐹 | |
5 | nfcv 2764 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑗 | |
6 | 4, 5 | nffv 6198 | . . . . . . 7 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
7 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑘ℂ | |
8 | 6, 7 | nfel 2777 | . . . . . 6 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ |
9 | fveq2 6191 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
10 | 9 | eleq1d 2686 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ)) |
11 | 3, 8, 10 | cbvral 3167 | . . . . 5 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ∈ ℂ) |
12 | 11 | biimpi 206 | . . . 4 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ∈ ℂ) |
13 | 12 | 3ad2ant3 1084 | . . 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 14402 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥) |
16 | 1, 2, 13, 15 | syl3anc 1326 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥) |
17 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑘abs | |
18 | 17, 6 | nffv 6198 | . . . . 5 ⊢ Ⅎ𝑘(abs‘(𝐹‘𝑗)) |
19 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑘 ≤ | |
20 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑘𝑥 | |
21 | 18, 19, 20 | nfbr 4699 | . . . 4 ⊢ Ⅎ𝑘(abs‘(𝐹‘𝑗)) ≤ 𝑥 |
22 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑘)) ≤ 𝑥 | |
23 | fveq2 6191 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘)) | |
24 | 23 | fveq2d 6195 | . . . . 5 ⊢ (𝑗 = 𝑘 → (abs‘(𝐹‘𝑗)) = (abs‘(𝐹‘𝑘))) |
25 | 24 | breq1d 4663 | . . . 4 ⊢ (𝑗 = 𝑘 → ((abs‘(𝐹‘𝑗)) ≤ 𝑥 ↔ (abs‘(𝐹‘𝑘)) ≤ 𝑥)) |
26 | 21, 22, 25 | cbvral 3167 | . . 3 ⊢ (∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
27 | 26 | rexbii 3041 | . 2 ⊢ (∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
28 | 16, 27 | sylib 208 | 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 1037 = wceq 1483 ∈ wcel 1990 Ⅎwnfc 2751 ∀wral 2912 ∃wrex 2913 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 ℂcc 9934 ℝcr 9935 ≤ cle 10075 ℤcz 11377 ℤ≥cuz 11687 abscabs 13974 ⇝ cli 14215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 |
This theorem is referenced by: climinf2mpt 39946 climinf3 39948 |
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