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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 12281 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11994 | . 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 25000 | . . 3 ⊢ (𝜑 → (𝐺‘𝑋):ℕ0⟶ℂ) |
7 | fvco3 6760 | . . 3 ⊢ (((𝐺‘𝑋):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((abs ∘ (𝐺‘𝑋))‘𝑘) = (abs‘((𝐺‘𝑋)‘𝑘))) | |
8 | 6, 7 | sylan 582 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((abs ∘ (𝐺‘𝑋))‘𝑘) = (abs‘((𝐺‘𝑋)‘𝑘))) |
9 | 6 | ffvelrnda 6851 | . 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 6675 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (abs‘((𝐺‘𝑋)‘𝑚)) = (abs‘((𝐺‘𝑋)‘𝑘))) | |
14 | 12, 13 | oveq12d 7174 | . . . . 5 ⊢ (𝑚 = 𝑘 → (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))) = (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘)))) |
15 | 14 | cbvmptv 5169 | . . . 4 ⊢ (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) = (𝑘 ∈ ℕ0 ↦ (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘)))) |
16 | 3, 4, 10, 5, 11, 15 | radcnvlt1 25006 | . . 3 ⊢ (𝜑 → (seq0( + , (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))))) ∈ dom ⇝ ∧ seq0( + , (abs ∘ (𝐺‘𝑋))) ∈ dom ⇝ )) |
17 | 16 | simprd 498 | . 2 ⊢ (𝜑 → seq0( + , (abs ∘ (𝐺‘𝑋))) ∈ dom ⇝ ) |
18 | 1, 2, 8, 9, 17 | abscvgcvg 15174 | 1 ⊢ (𝜑 → seq0( + , (𝐺‘𝑋)) ∈ dom ⇝ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3142 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 supcsup 8904 ℂcc 10535 ℝcr 10536 0cc0 10537 + caddc 10540 · cmul 10542 ℝ*cxr 10674 < clt 10675 ℕ0cn0 11898 seqcseq 13370 ↑cexp 13430 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-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 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 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 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-se 5515 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-isom 6364 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-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 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-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 |
This theorem is referenced by: pserulm 25010 pserdvlem2 25016 abelthlem3 25021 binomcxplemcvg 40706 |
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