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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 3650 | . . . . 5 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } ⊆ ℝ | |
3 | ressxr 9962 | . . . . 5 ⊢ ℝ ⊆ ℝ* | |
4 | 2, 3 | sstri 3577 | . . . 4 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ } ⊆ ℝ* |
5 | supxrcl 12017 | . . . 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 2692 | . 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 23974 | . . . 4 ⊢ (𝜑 → 0 ∈ {𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }) |
11 | supxrub 12026 | . . . 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 694 | . . 3 ⊢ (𝜑 → 0 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) |
13 | 12, 1 | syl6breqr 4625 | . 2 ⊢ (𝜑 → 0 ≤ 𝑅) |
14 | pnfge 11840 | . . 3 ⊢ (𝑅 ∈ ℝ* → 𝑅 ≤ +∞) | |
15 | 7, 14 | syl 17 | . 2 ⊢ (𝜑 → 𝑅 ≤ +∞) |
16 | 0xr 9965 | . . 3 ⊢ 0 ∈ ℝ* | |
17 | pnfxr 9971 | . . 3 ⊢ +∞ ∈ ℝ* | |
18 | elicc1 12090 | . . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑅 ∈ (0[,]+∞) ↔ (𝑅 ∈ ℝ* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞))) | |
19 | 16, 17, 18 | mp2an 704 | . 2 ⊢ (𝑅 ∈ (0[,]+∞) ↔ (𝑅 ∈ ℝ* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞)) |
20 | 7, 13, 15, 19 | syl3anbrc 1239 | 1 ⊢ (𝜑 → 𝑅 ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {crab 2900 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 supcsup 8229 ℂcc 9813 ℝcr 9814 0cc0 9815 + caddc 9818 · cmul 9820 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 ℕ0cn0 11169 [,]cicc 12049 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-icc 12053 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: radcnvlt1 23976 radcnvle 23978 pserulm 23980 psercnlem2 23982 psercnlem1 23983 psercn 23984 pserdvlem1 23985 pserdvlem2 23986 abelthlem3 23991 abelth 23999 logtayl 24206 |
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