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Mirrors > Home > MPE Home > Th. List > radcnv0 | Structured version Visualization version GIF version |
Description: Zero is always a convergent point for any power series. (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) |
Ref | Expression |
---|---|
radcnv0 | ⊢ (𝜑 → 0 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6645 | . . . 4 ⊢ (𝑟 = 0 → (𝐺‘𝑟) = (𝐺‘0)) | |
2 | 1 | seqeq3d 13372 | . . 3 ⊢ (𝑟 = 0 → seq0( + , (𝐺‘𝑟)) = seq0( + , (𝐺‘0))) |
3 | 2 | eleq1d 2874 | . 2 ⊢ (𝑟 = 0 → (seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ ↔ seq0( + , (𝐺‘0)) ∈ dom ⇝ )) |
4 | 0red 10633 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
5 | nn0uz 12268 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
6 | 0zd 11981 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
7 | snfi 8577 | . . . 4 ⊢ {0} ∈ Fin | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → {0} ∈ Fin) |
9 | 0nn0 11900 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℕ0) |
11 | 10 | snssd 4702 | . . 3 ⊢ (𝜑 → {0} ⊆ ℕ0) |
12 | ifid 4464 | . . . 4 ⊢ if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), ((𝐺‘0)‘𝑘)) = ((𝐺‘0)‘𝑘) | |
13 | 0cnd 10623 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℂ) | |
14 | pser.g | . . . . . . . . 9 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | |
15 | 14 | pserval2 25006 | . . . . . . . 8 ⊢ ((0 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) = ((𝐴‘𝑘) · (0↑𝑘))) |
16 | 13, 15 | sylan 583 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) = ((𝐴‘𝑘) · (0↑𝑘))) |
17 | 16 | adantr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐺‘0)‘𝑘) = ((𝐴‘𝑘) · (0↑𝑘))) |
18 | simpr 488 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
19 | elnn0 11887 | . . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0)) | |
20 | 18, 19 | sylib 221 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ ℕ ∨ 𝑘 = 0)) |
21 | 20 | ord 861 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ ℕ → 𝑘 = 0)) |
22 | velsn 4541 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0) | |
23 | 21, 22 | syl6ibr 255 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ ℕ → 𝑘 ∈ {0})) |
24 | 23 | con1d 147 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ {0} → 𝑘 ∈ ℕ)) |
25 | 24 | imp 410 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → 𝑘 ∈ ℕ) |
26 | 25 | 0expd 13499 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → (0↑𝑘) = 0) |
27 | 26 | oveq2d 7151 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐴‘𝑘) · (0↑𝑘)) = ((𝐴‘𝑘) · 0)) |
28 | radcnv.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
29 | 28 | ffvelrnda 6828 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
30 | 29 | adantr 484 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → (𝐴‘𝑘) ∈ ℂ) |
31 | 30 | mul01d 10828 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐴‘𝑘) · 0) = 0) |
32 | 17, 27, 31 | 3eqtrd 2837 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐺‘0)‘𝑘) = 0) |
33 | 32 | ifeq2da 4456 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), ((𝐺‘0)‘𝑘)) = if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), 0)) |
34 | 12, 33 | syl5eqr 2847 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) = if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), 0)) |
35 | 11 | sselda 3915 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {0}) → 𝑘 ∈ ℕ0) |
36 | 14, 28, 13 | psergf 25007 | . . . . 5 ⊢ (𝜑 → (𝐺‘0):ℕ0⟶ℂ) |
37 | 36 | ffvelrnda 6828 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) ∈ ℂ) |
38 | 35, 37 | syldan 594 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {0}) → ((𝐺‘0)‘𝑘) ∈ ℂ) |
39 | 5, 6, 8, 11, 34, 38 | fsumcvg3 15078 | . 2 ⊢ (𝜑 → seq0( + , (𝐺‘0)) ∈ dom ⇝ ) |
40 | 3, 4, 39 | elrabd 3630 | 1 ⊢ (𝜑 → 0 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 {crab 3110 ifcif 4425 {csn 4525 ↦ cmpt 5110 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 ℝcr 10525 0cc0 10526 + caddc 10529 · cmul 10531 ℕcn 11625 ℕ0cn0 11885 seqcseq 13364 ↑cexp 13425 ⇝ cli 14833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 |
This theorem is referenced by: radcnvcl 25012 radcnvrat 41018 |
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