Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6663 | . . . 4 ⊢ (𝑟 = 0 → (𝐺‘𝑟) = (𝐺‘0)) | |
2 | 1 | seqeq3d 13365 | . . 3 ⊢ (𝑟 = 0 → seq0( + , (𝐺‘𝑟)) = seq0( + , (𝐺‘0))) |
3 | 2 | eleq1d 2894 | . 2 ⊢ (𝑟 = 0 → (seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ ↔ seq0( + , (𝐺‘0)) ∈ dom ⇝ )) |
4 | 0red 10632 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
5 | nn0uz 12268 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
6 | 0zd 11981 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
7 | snfi 8582 | . . . 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 4734 | . . 3 ⊢ (𝜑 → {0} ⊆ ℕ0) |
12 | ifid 4502 | . . . 4 ⊢ if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), ((𝐺‘0)‘𝑘)) = ((𝐺‘0)‘𝑘) | |
13 | 0cnd 10622 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℂ) | |
14 | pser.g | . . . . . . . . 9 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | |
15 | 14 | pserval2 24926 | . . . . . . . 8 ⊢ ((0 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) = ((𝐴‘𝑘) · (0↑𝑘))) |
16 | 13, 15 | sylan 580 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) = ((𝐴‘𝑘) · (0↑𝑘))) |
17 | 16 | adantr 481 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐺‘0)‘𝑘) = ((𝐴‘𝑘) · (0↑𝑘))) |
18 | simpr 485 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
19 | elnn0 11887 | . . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0)) | |
20 | 18, 19 | sylib 219 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ ℕ ∨ 𝑘 = 0)) |
21 | 20 | ord 858 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ ℕ → 𝑘 = 0)) |
22 | velsn 4573 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0) | |
23 | 21, 22 | syl6ibr 253 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ ℕ → 𝑘 ∈ {0})) |
24 | 23 | con1d 147 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ {0} → 𝑘 ∈ ℕ)) |
25 | 24 | imp 407 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → 𝑘 ∈ ℕ) |
26 | 25 | 0expd 13491 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → (0↑𝑘) = 0) |
27 | 26 | oveq2d 7161 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐴‘𝑘) · (0↑𝑘)) = ((𝐴‘𝑘) · 0)) |
28 | radcnv.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
29 | 28 | ffvelrnda 6843 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
30 | 29 | adantr 481 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → (𝐴‘𝑘) ∈ ℂ) |
31 | 30 | mul01d 10827 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐴‘𝑘) · 0) = 0) |
32 | 17, 27, 31 | 3eqtrd 2857 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐺‘0)‘𝑘) = 0) |
33 | 32 | ifeq2da 4494 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), ((𝐺‘0)‘𝑘)) = if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), 0)) |
34 | 12, 33 | syl5eqr 2867 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) = if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), 0)) |
35 | 11 | sselda 3964 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {0}) → 𝑘 ∈ ℕ0) |
36 | 14, 28, 13 | psergf 24927 | . . . . 5 ⊢ (𝜑 → (𝐺‘0):ℕ0⟶ℂ) |
37 | 36 | ffvelrnda 6843 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) ∈ ℂ) |
38 | 35, 37 | syldan 591 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {0}) → ((𝐺‘0)‘𝑘) ∈ ℂ) |
39 | 5, 6, 8, 11, 34, 38 | fsumcvg3 15074 | . 2 ⊢ (𝜑 → seq0( + , (𝐺‘0)) ∈ dom ⇝ ) |
40 | 3, 4, 39 | elrabd 3679 | 1 ⊢ (𝜑 → 0 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 {crab 3139 ifcif 4463 {csn 4557 ↦ cmpt 5137 dom cdm 5548 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 ℂcc 10523 ℝcr 10524 0cc0 10525 + caddc 10528 · cmul 10530 ℕcn 11626 ℕ0cn0 11885 seqcseq 13357 ↑cexp 13417 ⇝ cli 14829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 |
This theorem is referenced by: radcnvcl 24932 radcnvrat 40523 |
Copyright terms: Public domain | W3C validator |