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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 | 0red 9920 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
2 | nn0uz 11598 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 0zd 11266 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
4 | snfi 7923 | . . . 4 ⊢ {0} ∈ Fin | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → {0} ∈ Fin) |
6 | 0nn0 11184 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℕ0) |
8 | 7 | snssd 4281 | . . 3 ⊢ (𝜑 → {0} ⊆ ℕ0) |
9 | ifid 4075 | . . . 4 ⊢ if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), ((𝐺‘0)‘𝑘)) = ((𝐺‘0)‘𝑘) | |
10 | 0cnd 9912 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℂ) | |
11 | pser.g | . . . . . . . . 9 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | |
12 | 11 | pserval2 23969 | . . . . . . . 8 ⊢ ((0 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) = ((𝐴‘𝑘) · (0↑𝑘))) |
13 | 10, 12 | sylan 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) = ((𝐴‘𝑘) · (0↑𝑘))) |
14 | 13 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐺‘0)‘𝑘) = ((𝐴‘𝑘) · (0↑𝑘))) |
15 | simpr 476 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
16 | elnn0 11171 | . . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0)) | |
17 | 15, 16 | sylib 207 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ ℕ ∨ 𝑘 = 0)) |
18 | 17 | ord 391 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ ℕ → 𝑘 = 0)) |
19 | velsn 4141 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0) | |
20 | 18, 19 | syl6ibr 241 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ ℕ → 𝑘 ∈ {0})) |
21 | 20 | con1d 138 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ {0} → 𝑘 ∈ ℕ)) |
22 | 21 | imp 444 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → 𝑘 ∈ ℕ) |
23 | 22 | 0expd 12886 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → (0↑𝑘) = 0) |
24 | 23 | oveq2d 6565 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐴‘𝑘) · (0↑𝑘)) = ((𝐴‘𝑘) · 0)) |
25 | radcnv.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
26 | 25 | ffvelrnda 6267 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
27 | 26 | adantr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → (𝐴‘𝑘) ∈ ℂ) |
28 | 27 | mul01d 10114 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐴‘𝑘) · 0) = 0) |
29 | 14, 24, 28 | 3eqtrd 2648 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐺‘0)‘𝑘) = 0) |
30 | 29 | ifeq2da 4067 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), ((𝐺‘0)‘𝑘)) = if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), 0)) |
31 | 9, 30 | syl5eqr 2658 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) = if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), 0)) |
32 | 8 | sselda 3568 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {0}) → 𝑘 ∈ ℕ0) |
33 | 11, 25, 10 | psergf 23970 | . . . . 5 ⊢ (𝜑 → (𝐺‘0):ℕ0⟶ℂ) |
34 | 33 | ffvelrnda 6267 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) ∈ ℂ) |
35 | 32, 34 | syldan 486 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {0}) → ((𝐺‘0)‘𝑘) ∈ ℂ) |
36 | 2, 3, 5, 8, 31, 35 | fsumcvg3 14307 | . 2 ⊢ (𝜑 → seq0( + , (𝐺‘0)) ∈ dom ⇝ ) |
37 | fveq2 6103 | . . . . 5 ⊢ (𝑟 = 0 → (𝐺‘𝑟) = (𝐺‘0)) | |
38 | 37 | seqeq3d 12671 | . . . 4 ⊢ (𝑟 = 0 → seq0( + , (𝐺‘𝑟)) = seq0( + , (𝐺‘0))) |
39 | 38 | eleq1d 2672 | . . 3 ⊢ (𝑟 = 0 → (seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ ↔ seq0( + , (𝐺‘0)) ∈ dom ⇝ )) |
40 | 39 | elrab 3331 | . 2 ⊢ (0 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } ↔ (0 ∈ ℝ ∧ seq0( + , (𝐺‘0)) ∈ dom ⇝ )) |
41 | 1, 36, 40 | sylanbrc 695 | 1 ⊢ (𝜑 → 0 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ifcif 4036 {csn 4125 ↦ cmpt 4643 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 ℝcr 9814 0cc0 9815 + caddc 9818 · cmul 9820 ℕcn 10897 ℕ0cn0 11169 seqcseq 12663 ↑cexp 12722 ⇝ cli 14063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 |
This theorem is referenced by: radcnvcl 23975 radcnvrat 37535 |
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