| 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 1137 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → 𝑀 ∈ ℤ) | |
| 2 | simp2 1138 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → 𝐹 ∈ dom ⇝ ) | |
| 3 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑗(𝐹‘𝑘) ∈ ℂ | |
| 4 | climbddf.1 | . . . . . . . 8 ⊢ Ⅎ𝑘𝐹 | |
| 5 | nfcv 2905 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑗 | |
| 6 | 4, 5 | nffv 6916 | . . . . . . 7 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
| 7 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑘ℂ | |
| 8 | 6, 7 | nfel 2920 | . . . . . 6 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ |
| 9 | fveq2 6906 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
| 10 | 9 | eleq1d 2826 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ)) |
| 11 | 3, 8, 10 | cbvralw 3306 | . . . . 5 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ∈ ℂ) |
| 12 | 11 | biimpi 216 | . . . 4 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ∈ ℂ) |
| 13 | 12 | 3ad2ant3 1136 | . . 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 15708 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥) |
| 16 | 1, 2, 13, 15 | syl3anc 1373 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥) |
| 17 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑘abs | |
| 18 | 17, 6 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑘(abs‘(𝐹‘𝑗)) |
| 19 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑘 ≤ | |
| 20 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑘𝑥 | |
| 21 | 18, 19, 20 | nfbr 5190 | . . . 4 ⊢ Ⅎ𝑘(abs‘(𝐹‘𝑗)) ≤ 𝑥 |
| 22 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑗(abs‘(𝐹‘𝑘)) ≤ 𝑥 | |
| 23 | 2fveq3 6911 | . . . . 5 ⊢ (𝑗 = 𝑘 → (abs‘(𝐹‘𝑗)) = (abs‘(𝐹‘𝑘))) | |
| 24 | 23 | breq1d 5153 | . . . 4 ⊢ (𝑗 = 𝑘 → ((abs‘(𝐹‘𝑗)) ≤ 𝑥 ↔ (abs‘(𝐹‘𝑘)) ≤ 𝑥)) |
| 25 | 21, 22, 24 | cbvralw 3306 | . . 3 ⊢ (∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
| 26 | 25 | rexbii 3094 | . 2 ⊢ (∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (abs‘(𝐹‘𝑗)) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) |
| 27 | 16, 26 | sylib 218 | 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 1087 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 ∃wrex 3070 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 ℂcc 11153 ℝcr 11154 ≤ cle 11296 ℤcz 12613 ℤ≥cuz 12878 abscabs 15273 ⇝ cli 15520 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 |
| This theorem is referenced by: climinf2mpt 45729 climinf3 45731 |
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