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Mirrors > Home > MPE Home > Th. List > radcnvcl | Structured version Visualization version GIF version |
Description: The radius of convergence 𝑅 of an infinite series is a nonnegative extended real number. (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) |
radcnv.r | ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) |
Ref | Expression |
---|---|
radcnvcl | ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | radcnv.r | . . 3 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) | |
2 | ssrab2 3914 | . . . . 5 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } ⊆ ℝ | |
3 | ressxr 10407 | . . . . 5 ⊢ ℝ ⊆ ℝ* | |
4 | 2, 3 | sstri 3836 | . . . 4 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } ⊆ ℝ* |
5 | supxrcl 12440 | . . . 4 ⊢ ({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } ⊆ ℝ* → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*) | |
6 | 4, 5 | mp1i 13 | . . 3 ⊢ (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*) |
7 | 1, 6 | syl5eqel 2910 | . 2 ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
8 | pser.g | . . . . 5 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | |
9 | radcnv.a | . . . . 5 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
10 | 8, 9 | radcnv0 24576 | . . . 4 ⊢ (𝜑 → 0 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
11 | supxrub 12449 | . . . 4 ⊢ (({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } ⊆ ℝ* ∧ 0 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) → 0 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) | |
12 | 4, 10, 11 | sylancr 581 | . . 3 ⊢ (𝜑 → 0 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) |
13 | 12, 1 | syl6breqr 4917 | . 2 ⊢ (𝜑 → 0 ≤ 𝑅) |
14 | pnfge 12257 | . . 3 ⊢ (𝑅 ∈ ℝ* → 𝑅 ≤ +∞) | |
15 | 7, 14 | syl 17 | . 2 ⊢ (𝜑 → 𝑅 ≤ +∞) |
16 | 0xr 10410 | . . 3 ⊢ 0 ∈ ℝ* | |
17 | pnfxr 10417 | . . 3 ⊢ +∞ ∈ ℝ* | |
18 | elicc1 12514 | . . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑅 ∈ (0[,]+∞) ↔ (𝑅 ∈ ℝ* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞))) | |
19 | 16, 17, 18 | mp2an 683 | . 2 ⊢ (𝑅 ∈ (0[,]+∞) ↔ (𝑅 ∈ ℝ* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞)) |
20 | 7, 13, 15, 19 | syl3anbrc 1447 | 1 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 {crab 3121 ⊆ wss 3798 class class class wbr 4875 ↦ cmpt 4954 dom cdm 5346 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 supcsup 8621 ℂcc 10257 ℝcr 10258 0cc0 10259 + caddc 10262 · cmul 10264 +∞cpnf 10395 ℝ*cxr 10397 < clt 10398 ≤ cle 10399 ℕ0cn0 11625 [,]cicc 12473 seqcseq 13102 ↑cexp 13161 ⇝ cli 14599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-n0 11626 df-z 11712 df-uz 11976 df-rp 12120 df-icc 12477 df-fz 12627 df-seq 13103 df-exp 13162 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-clim 14603 |
This theorem is referenced by: radcnvlt1 24578 radcnvle 24580 pserulm 24582 psercnlem2 24584 psercnlem1 24585 psercn 24586 pserdvlem1 24587 pserdvlem2 24588 abelthlem3 24593 abelth 24601 logtayl 24812 |
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