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Mirrors > Home > MPE Home > Th. List > radcnvlt2 | Structured version Visualization version GIF version |
Description: If 𝑋 is within the open disk of radius 𝑅 centered at zero, then the infinite series converges at 𝑋. (Contributed by Mario Carneiro, 26-Feb-2015.) |
Ref | Expression |
---|---|
pser.g | ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) |
radcnv.a | ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
radcnv.r | ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) |
radcnvlt.x | ⊢ (𝜑 → 𝑋 ∈ ℂ) |
radcnvlt.a | ⊢ (𝜑 → (abs‘𝑋) < 𝑅) |
Ref | Expression |
---|---|
radcnvlt2 | ⊢ (𝜑 → seq0( + , (𝐺‘𝑋)) ∈ dom ⇝ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12897 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 12603 | . 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 | radcnvlt.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ) | |
6 | 3, 4, 5 | psergf 26393 | . . 3 ⊢ (𝜑 → (𝐺‘𝑋):ℕ0⟶ℂ) |
7 | fvco3 6996 | . . 3 ⊢ (((𝐺‘𝑋):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((abs ∘ (𝐺‘𝑋))‘𝑘) = (abs‘((𝐺‘𝑋)‘𝑘))) | |
8 | 6, 7 | sylan 578 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((abs ∘ (𝐺‘𝑋))‘𝑘) = (abs‘((𝐺‘𝑋)‘𝑘))) |
9 | 6 | ffvelcdmda 7093 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘𝑋)‘𝑘) ∈ ℂ) |
10 | radcnv.r | . . . 4 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) | |
11 | radcnvlt.a | . . . 4 ⊢ (𝜑 → (abs‘𝑋) < 𝑅) | |
12 | id 22 | . . . . . 6 ⊢ (𝑚 = 𝑘 → 𝑚 = 𝑘) | |
13 | 2fveq3 6901 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (abs‘((𝐺‘𝑋)‘𝑚)) = (abs‘((𝐺‘𝑋)‘𝑘))) | |
14 | 12, 13 | oveq12d 7437 | . . . . 5 ⊢ (𝑚 = 𝑘 → (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))) = (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘)))) |
15 | 14 | cbvmptv 5262 | . . . 4 ⊢ (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) = (𝑘 ∈ ℕ0 ↦ (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘)))) |
16 | 3, 4, 10, 5, 11, 15 | radcnvlt1 26399 | . . 3 ⊢ (𝜑 → (seq0( + , (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))))) ∈ dom ⇝ ∧ seq0( + , (abs ∘ (𝐺‘𝑋))) ∈ dom ⇝ )) |
17 | 16 | simprd 494 | . 2 ⊢ (𝜑 → seq0( + , (abs ∘ (𝐺‘𝑋))) ∈ dom ⇝ ) |
18 | 1, 2, 8, 9, 17 | abscvgcvg 15801 | 1 ⊢ (𝜑 → seq0( + , (𝐺‘𝑋)) ∈ dom ⇝ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3418 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5678 ∘ ccom 5682 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 supcsup 9465 ℂcc 11138 ℝcr 11139 0cc0 11140 + caddc 11143 · cmul 11145 ℝ*cxr 11279 < clt 11280 ℕ0cn0 12505 seqcseq 14002 ↑cexp 14062 abscabs 15217 ⇝ cli 15464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-limsup 15451 df-clim 15468 df-rlim 15469 df-sum 15669 |
This theorem is referenced by: pserulm 26403 pserdvlem2 26410 abelthlem3 26415 binomcxplemcvg 43933 |
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