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Mirrors > Home > MPE Home > Th. List > radcnvlem3 | Structured version Visualization version GIF version |
Description: Lemma for radcnvlt1 25912, radcnvle 25914. If 𝑋 is a point closer to zero than 𝑌 and the power series converges at 𝑌, then it converges at 𝑋. (Contributed by Mario Carneiro, 31-Mar-2015.) |
Ref | Expression |
---|---|
pser.g | ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) |
radcnv.a | ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
psergf.x | ⊢ (𝜑 → 𝑋 ∈ ℂ) |
radcnvlem2.y | ⊢ (𝜑 → 𝑌 ∈ ℂ) |
radcnvlem2.a | ⊢ (𝜑 → (abs‘𝑋) < (abs‘𝑌)) |
radcnvlem2.c | ⊢ (𝜑 → seq0( + , (𝐺‘𝑌)) ∈ dom ⇝ ) |
Ref | Expression |
---|---|
radcnvlem3 | ⊢ (𝜑 → seq0( + , (𝐺‘𝑋)) ∈ dom ⇝ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12860 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 12566 | . 2 ⊢ (𝜑 → 0 ∈ ℤ) | |
3 | pser.g | . . . 4 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | |
4 | radcnv.a | . . . 4 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
5 | psergf.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ) | |
6 | 3, 4, 5 | psergf 25906 | . . 3 ⊢ (𝜑 → (𝐺‘𝑋):ℕ0⟶ℂ) |
7 | fvco3 6986 | . . 3 ⊢ (((𝐺‘𝑋):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((abs ∘ (𝐺‘𝑋))‘𝑘) = (abs‘((𝐺‘𝑋)‘𝑘))) | |
8 | 6, 7 | sylan 581 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((abs ∘ (𝐺‘𝑋))‘𝑘) = (abs‘((𝐺‘𝑋)‘𝑘))) |
9 | 6 | ffvelcdmda 7082 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘𝑋)‘𝑘) ∈ ℂ) |
10 | radcnvlem2.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ ℂ) | |
11 | radcnvlem2.a | . . 3 ⊢ (𝜑 → (abs‘𝑋) < (abs‘𝑌)) | |
12 | radcnvlem2.c | . . 3 ⊢ (𝜑 → seq0( + , (𝐺‘𝑌)) ∈ dom ⇝ ) | |
13 | 3, 4, 5, 10, 11, 12 | radcnvlem2 25908 | . 2 ⊢ (𝜑 → seq0( + , (abs ∘ (𝐺‘𝑋))) ∈ dom ⇝ ) |
14 | 1, 2, 8, 9, 13 | abscvgcvg 15761 | 1 ⊢ (𝜑 → seq0( + , (𝐺‘𝑋)) ∈ dom ⇝ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ∘ ccom 5679 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ℂcc 11104 0cc0 11106 + caddc 11109 · cmul 11111 < clt 11244 ℕ0cn0 12468 seqcseq 13962 ↑cexp 14023 abscabs 15177 ⇝ cli 15424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-ico 13326 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 |
This theorem is referenced by: radcnvle 25914 |
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