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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 6907 | . . . 4 ⊢ (𝑟 = 0 → (𝐺‘𝑟) = (𝐺‘0)) | |
2 | 1 | seqeq3d 14047 | . . 3 ⊢ (𝑟 = 0 → seq0( + , (𝐺‘𝑟)) = seq0( + , (𝐺‘0))) |
3 | 2 | eleq1d 2824 | . 2 ⊢ (𝑟 = 0 → (seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ ↔ seq0( + , (𝐺‘0)) ∈ dom ⇝ )) |
4 | 0red 11262 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
5 | nn0uz 12918 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
6 | 0zd 12623 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
7 | snfi 9082 | . . . 4 ⊢ {0} ∈ Fin | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → {0} ∈ Fin) |
9 | 0nn0 12539 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℕ0) |
11 | 10 | snssd 4814 | . . 3 ⊢ (𝜑 → {0} ⊆ ℕ0) |
12 | ifid 4571 | . . . 4 ⊢ if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), ((𝐺‘0)‘𝑘)) = ((𝐺‘0)‘𝑘) | |
13 | 0cnd 11252 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℂ) | |
14 | pser.g | . . . . . . . . 9 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | |
15 | 14 | pserval2 26469 | . . . . . . . 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 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐺‘0)‘𝑘) = ((𝐴‘𝑘) · (0↑𝑘))) |
18 | simpr 484 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
19 | elnn0 12526 | . . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0)) | |
20 | 18, 19 | sylib 218 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ ℕ ∨ 𝑘 = 0)) |
21 | 20 | ord 864 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ ℕ → 𝑘 = 0)) |
22 | velsn 4647 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0) | |
23 | 21, 22 | imbitrrdi 252 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ ℕ → 𝑘 ∈ {0})) |
24 | 23 | con1d 145 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ 𝑘 ∈ {0} → 𝑘 ∈ ℕ)) |
25 | 24 | imp 406 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → 𝑘 ∈ ℕ) |
26 | 25 | 0expd 14176 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → (0↑𝑘) = 0) |
27 | 26 | oveq2d 7447 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐴‘𝑘) · (0↑𝑘)) = ((𝐴‘𝑘) · 0)) |
28 | radcnv.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
29 | 28 | ffvelcdmda 7104 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
30 | 29 | adantr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → (𝐴‘𝑘) ∈ ℂ) |
31 | 30 | mul01d 11458 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐴‘𝑘) · 0) = 0) |
32 | 17, 27, 31 | 3eqtrd 2779 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ ¬ 𝑘 ∈ {0}) → ((𝐺‘0)‘𝑘) = 0) |
33 | 32 | ifeq2da 4563 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), ((𝐺‘0)‘𝑘)) = if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), 0)) |
34 | 12, 33 | eqtr3id 2789 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐺‘0)‘𝑘) = if(𝑘 ∈ {0}, ((𝐺‘0)‘𝑘), 0)) |
35 | 11 | sselda 3995 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {0}) → 𝑘 ∈ ℕ0) |
36 | 14, 28, 13 | psergf 26470 | . . . . 5 ⊢ (𝜑 → (𝐺‘0):ℕ0⟶ℂ) |
37 | 36 | ffvelcdmda 7104 | . . . 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 15762 | . 2 ⊢ (𝜑 → seq0( + , (𝐺‘0)) ∈ dom ⇝ ) |
40 | 3, 4, 39 | elrabd 3697 | 1 ⊢ (𝜑 → 0 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 {crab 3433 ifcif 4531 {csn 4631 ↦ cmpt 5231 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℂcc 11151 ℝcr 11152 0cc0 11153 + caddc 11156 · cmul 11158 ℕcn 12264 ℕ0cn0 12524 seqcseq 14039 ↑cexp 14099 ⇝ cli 15517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 |
This theorem is referenced by: radcnvcl 26475 radcnvrat 44310 |
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