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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 6842 | . . . 4 ⊢ (𝑟 = 0 → (𝐺‘𝑟) = (𝐺‘0)) | |
2 | 1 | seqeq3d 13914 | . . 3 ⊢ (𝑟 = 0 → seq0( + , (𝐺‘𝑟)) = seq0( + , (𝐺‘0))) |
3 | 2 | eleq1d 2822 | . 2 ⊢ (𝑟 = 0 → (seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ ↔ seq0( + , (𝐺‘0)) ∈ dom ⇝ )) |
4 | 0red 11158 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
5 | nn0uz 12805 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
6 | 0zd 12511 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
7 | snfi 8988 | . . . 4 ⊢ {0} ∈ Fin | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → {0} ∈ Fin) |
9 | 0nn0 12428 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℕ0) |
11 | 10 | snssd 4769 | . . 3 ⊢ (𝜑 → {0} ⊆ ℕ0) |
12 | ifid 4526 | . . . 4 ⊢ if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), ((𝐺‘0)‘𝑘)) = ((𝐺‘0)‘𝑘) | |
13 | 0cnd 11148 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℂ) | |
14 | pser.g | . . . . . . . . 9 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | |
15 | 14 | pserval2 25770 | . . . . . . . 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 12415 | . . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0)) | |
20 | 18, 19 | sylib 217 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ ℕ ∨ 𝑘 = 0)) |
21 | 20 | ord 862 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ ℕ → 𝑘 = 0)) |
22 | velsn 4602 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0) | |
23 | 21, 22 | syl6ibr 251 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ ℕ → 𝑘 ∈ {0})) |
24 | 23 | con1d 145 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ {0} → 𝑘 ∈ ℕ)) |
25 | 24 | imp 407 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → 𝑘 ∈ ℕ) |
26 | 25 | 0expd 14044 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → (0↑𝑘) = 0) |
27 | 26 | oveq2d 7373 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐴‘𝑘) · (0↑𝑘)) = ((𝐴‘𝑘) · 0)) |
28 | radcnv.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
29 | 28 | ffvelcdmda 7035 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
30 | 29 | adantr 481 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → (𝐴‘𝑘) ∈ ℂ) |
31 | 30 | mul01d 11354 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐴‘𝑘) · 0) = 0) |
32 | 17, 27, 31 | 3eqtrd 2780 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐺‘0)‘𝑘) = 0) |
33 | 32 | ifeq2da 4518 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), ((𝐺‘0)‘𝑘)) = if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), 0)) |
34 | 12, 33 | eqtr3id 2790 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) = if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), 0)) |
35 | 11 | sselda 3944 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {0}) → 𝑘 ∈ ℕ0) |
36 | 14, 28, 13 | psergf 25771 | . . . . 5 ⊢ (𝜑 → (𝐺‘0):ℕ0⟶ℂ) |
37 | 36 | ffvelcdmda 7035 | . . . 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 15614 | . 2 ⊢ (𝜑 → seq0( + , (𝐺‘0)) ∈ dom ⇝ ) |
40 | 3, 4, 39 | elrabd 3647 | 1 ⊢ (𝜑 → 0 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 {crab 3407 ifcif 4486 {csn 4586 ↦ cmpt 5188 dom cdm 5633 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℂcc 11049 ℝcr 11050 0cc0 11051 + caddc 11054 · cmul 11056 ℕcn 12153 ℕ0cn0 12413 seqcseq 13906 ↑cexp 13967 ⇝ cli 15366 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 |
This theorem is referenced by: radcnvcl 25776 radcnvrat 42584 |
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