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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 12283 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11996 | . 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 25002 | . . 3 ⊢ (𝜑 → (𝐺‘𝑋):ℕ0⟶ℂ) |
7 | fvco3 6762 | . . 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 6853 | . 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 6677 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (abs‘((𝐺‘𝑋)‘𝑚)) = (abs‘((𝐺‘𝑋)‘𝑘))) | |
14 | 12, 13 | oveq12d 7176 | . . . . 5 ⊢ (𝑚 = 𝑘 → (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚))) = (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘)))) |
15 | 14 | cbvmptv 5171 | . . . 4 ⊢ (𝑚 ∈ ℕ0 ↦ (𝑚 · (abs‘((𝐺‘𝑋)‘𝑚)))) = (𝑘 ∈ ℕ0 ↦ (𝑘 · (abs‘((𝐺‘𝑋)‘𝑘)))) |
16 | 3, 4, 10, 5, 11, 15 | radcnvlt1 25008 | . . 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 15176 | 1 ⊢ (𝜑 → seq0( + , (𝐺‘𝑋)) ∈ dom ⇝ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3144 class class class wbr 5068 ↦ cmpt 5148 dom cdm 5557 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 supcsup 8906 ℂcc 10537 ℝcr 10538 0cc0 10539 + caddc 10542 · cmul 10544 ℝ*cxr 10676 < clt 10677 ℕ0cn0 11900 seqcseq 13372 ↑cexp 13432 abscabs 14595 ⇝ cli 14843 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 |
This theorem is referenced by: pserulm 25012 pserdvlem2 25018 abelthlem3 25023 binomcxplemcvg 40693 |
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