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 1132 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → 𝑀 ∈ ℤ) | |
2 | simp2 1133 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → 𝐹 ∈ dom ⇝ ) | |
3 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑗(𝐹‘𝑘) ∈ ℂ | |
4 | climbddf.1 | . . . . . . . 8 ⊢ Ⅎ𝑘𝐹 | |
5 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑗 | |
6 | 4, 5 | nffv 6680 | . . . . . . 7 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
7 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑘ℂ | |
8 | 6, 7 | nfel 2992 | . . . . . 6 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ |
9 | fveq2 6670 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
10 | 9 | eleq1d 2897 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ)) |
11 | 3, 8, 10 | cbvralw 3441 | . . . . 5 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ∈ ℂ) |
12 | 11 | biimpi 218 | . . . 4 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ∈ ℂ) |
13 | 12 | 3ad2ant3 1131 | . . 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 15028 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥) |
16 | 1, 2, 13, 15 | syl3anc 1367 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥) |
17 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑘abs | |
18 | 17, 6 | nffv 6680 | . . . . 5 ⊢ Ⅎ𝑘(abs‘(𝐹‘𝑗)) |
19 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑘 ≤ | |
20 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑘𝑥 | |
21 | 18, 19, 20 | nfbr 5113 | . . . 4 ⊢ Ⅎ𝑘(abs‘(𝐹‘𝑗)) ≤ 𝑥 |
22 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑘)) ≤ 𝑥 | |
23 | 2fveq3 6675 | . . . . 5 ⊢ (𝑗 = 𝑘 → (abs‘(𝐹‘𝑗)) = (abs‘(𝐹‘𝑘))) | |
24 | 23 | breq1d 5076 | . . . 4 ⊢ (𝑗 = 𝑘 → ((abs‘(𝐹‘𝑗)) ≤ 𝑥 ↔ (abs‘(𝐹‘𝑘)) ≤ 𝑥)) |
25 | 21, 22, 24 | cbvralw 3441 | . . 3 ⊢ (∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
26 | 25 | rexbii 3247 | . 2 ⊢ (∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
27 | 16, 26 | sylib 220 | 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 1083 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2961 ∀wral 3138 ∃wrex 3139 class class class wbr 5066 dom cdm 5555 ‘cfv 6355 ℂcc 10535 ℝcr 10536 ≤ cle 10676 ℤcz 11982 ℤ≥cuz 12244 abscabs 14593 ⇝ cli 14841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 |
This theorem is referenced by: climinf2mpt 42015 climinf3 42017 |
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