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Mirrors > Home > MPE Home > Th. List > Mathboxes > binomcxplemradcnv | Structured version Visualization version GIF version |
Description: Lemma for binomcxp 41061. By binomcxplemfrat 41055 and radcnvrat 41018 the radius of convergence of power series Σ𝑘 ∈ ℕ0((𝐹‘𝑘) · (𝑏↑𝑘)) is one. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
Ref | Expression |
---|---|
binomcxp.a | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
binomcxp.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
binomcxp.lt | ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴)) |
binomcxp.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
binomcxplem.f | ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗)) |
binomcxplem.s | ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘)))) |
binomcxplem.r | ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) |
Ref | Expression |
---|---|
binomcxplemradcnv | ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → 𝑅 = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | binomcxplem.s | . . . 4 ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘)))) | |
2 | simpl 486 | . . . . . . . . 9 ⊢ ((𝑏 = 𝑥 ∧ 𝑘 ∈ ℕ0) → 𝑏 = 𝑥) | |
3 | 2 | oveq1d 7150 | . . . . . . . 8 ⊢ ((𝑏 = 𝑥 ∧ 𝑘 ∈ ℕ0) → (𝑏↑𝑘) = (𝑥↑𝑘)) |
4 | 3 | oveq2d 7151 | . . . . . . 7 ⊢ ((𝑏 = 𝑥 ∧ 𝑘 ∈ ℕ0) → ((𝐹‘𝑘) · (𝑏↑𝑘)) = ((𝐹‘𝑘) · (𝑥↑𝑘))) |
5 | 4 | mpteq2dva 5125 | . . . . . 6 ⊢ (𝑏 = 𝑥 → (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))) = (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑥↑𝑘)))) |
6 | fveq2 6645 | . . . . . . . 8 ⊢ (𝑘 = 𝑦 → (𝐹‘𝑘) = (𝐹‘𝑦)) | |
7 | oveq2 7143 | . . . . . . . 8 ⊢ (𝑘 = 𝑦 → (𝑥↑𝑘) = (𝑥↑𝑦)) | |
8 | 6, 7 | oveq12d 7153 | . . . . . . 7 ⊢ (𝑘 = 𝑦 → ((𝐹‘𝑘) · (𝑥↑𝑘)) = ((𝐹‘𝑦) · (𝑥↑𝑦))) |
9 | 8 | cbvmptv 5133 | . . . . . 6 ⊢ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑥↑𝑘))) = (𝑦 ∈ ℕ0 ↦ ((𝐹‘𝑦) · (𝑥↑𝑦))) |
10 | 5, 9 | eqtrdi 2849 | . . . . 5 ⊢ (𝑏 = 𝑥 → (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))) = (𝑦 ∈ ℕ0 ↦ ((𝐹‘𝑦) · (𝑥↑𝑦)))) |
11 | 10 | cbvmptv 5133 | . . . 4 ⊢ (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘)))) = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℕ0 ↦ ((𝐹‘𝑦) · (𝑥↑𝑦)))) |
12 | 1, 11 | eqtri 2821 | . . 3 ⊢ 𝑆 = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℕ0 ↦ ((𝐹‘𝑦) · (𝑥↑𝑦)))) |
13 | binomcxp.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
14 | 13 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) → 𝐶 ∈ ℂ) |
15 | simpr 488 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) → 𝑗 ∈ ℕ0) | |
16 | 14, 15 | bcccl 41043 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) → (𝐶C𝑐𝑗) ∈ ℂ) |
17 | binomcxplem.f | . . . 4 ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗)) | |
18 | 16, 17 | fmptd 6855 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → 𝐹:ℕ0⟶ℂ) |
19 | binomcxplem.r | . . 3 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) | |
20 | fvoveq1 7158 | . . . . . 6 ⊢ (𝑘 = 𝑖 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑖 + 1))) | |
21 | fveq2 6645 | . . . . . 6 ⊢ (𝑘 = 𝑖 → (𝐹‘𝑘) = (𝐹‘𝑖)) | |
22 | 20, 21 | oveq12d 7153 | . . . . 5 ⊢ (𝑘 = 𝑖 → ((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘)) = ((𝐹‘(𝑖 + 1)) / (𝐹‘𝑖))) |
23 | 22 | fveq2d 6649 | . . . 4 ⊢ (𝑘 = 𝑖 → (abs‘((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘))) = (abs‘((𝐹‘(𝑖 + 1)) / (𝐹‘𝑖)))) |
24 | 23 | cbvmptv 5133 | . . 3 ⊢ (𝑘 ∈ ℕ0 ↦ (abs‘((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘)))) = (𝑖 ∈ ℕ0 ↦ (abs‘((𝐹‘(𝑖 + 1)) / (𝐹‘𝑖)))) |
25 | nn0uz 12268 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
26 | 0nn0 11900 | . . . 4 ⊢ 0 ∈ ℕ0 | |
27 | 26 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → 0 ∈ ℕ0) |
28 | 17 | a1i 11 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗))) |
29 | simpr 488 | . . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 = 𝑖) → 𝑗 = 𝑖) | |
30 | 29 | oveq2d 7151 | . . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 = 𝑖) → (𝐶C𝑐𝑗) = (𝐶C𝑐𝑖)) |
31 | simpr 488 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0) | |
32 | ovexd 7170 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → (𝐶C𝑐𝑖) ∈ V) | |
33 | 28, 30, 31, 32 | fvmptd 6752 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → (𝐹‘𝑖) = (𝐶C𝑐𝑖)) |
34 | elfznn0 12995 | . . . . . . 7 ⊢ (𝐶 ∈ (0...(𝑖 − 1)) → 𝐶 ∈ ℕ0) | |
35 | 34 | con3i 157 | . . . . . 6 ⊢ (¬ 𝐶 ∈ ℕ0 → ¬ 𝐶 ∈ (0...(𝑖 − 1))) |
36 | 35 | ad2antlr 726 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → ¬ 𝐶 ∈ (0...(𝑖 − 1))) |
37 | 13 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐶 ∈ ℂ) |
38 | simpr 488 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0) | |
39 | 37, 38 | bcc0 41044 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝐶C𝑐𝑖) = 0 ↔ 𝐶 ∈ (0...(𝑖 − 1)))) |
40 | 39 | necon3abid 3023 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝐶C𝑐𝑖) ≠ 0 ↔ ¬ 𝐶 ∈ (0...(𝑖 − 1)))) |
41 | 40 | adantlr 714 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → ((𝐶C𝑐𝑖) ≠ 0 ↔ ¬ 𝐶 ∈ (0...(𝑖 − 1)))) |
42 | 36, 41 | mpbird 260 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → (𝐶C𝑐𝑖) ≠ 0) |
43 | 33, 42 | eqnetrd 3054 | . . 3 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → (𝐹‘𝑖) ≠ 0) |
44 | binomcxp.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
45 | binomcxp.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
46 | binomcxp.lt | . . . 4 ⊢ (𝜑 → (abs‘𝐵) < (abs‘𝐴)) | |
47 | 44, 45, 46, 13, 17 | binomcxplemfrat 41055 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ (abs‘((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘)))) ⇝ 1) |
48 | ax-1ne0 10595 | . . . 4 ⊢ 1 ≠ 0 | |
49 | 48 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → 1 ≠ 0) |
50 | 12, 18, 19, 24, 25, 27, 43, 47, 49 | radcnvrat 41018 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → 𝑅 = (1 / 1)) |
51 | 1div1e1 11319 | . 2 ⊢ (1 / 1) = 1 | |
52 | 50, 51 | eqtrdi 2849 | 1 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → 𝑅 = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {crab 3110 Vcvv 3441 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 supcsup 8888 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ℝ*cxr 10663 < clt 10664 − cmin 10859 / cdiv 11286 ℕ0cn0 11885 ℝ+crp 12377 ...cfz 12885 seqcseq 13364 ↑cexp 13425 abscabs 14585 ⇝ cli 14833 C𝑐cbcc 41040 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-ioo 12730 df-ico 12732 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-fac 13630 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-prod 15252 df-fallfac 15353 df-bcc 41041 |
This theorem is referenced by: binomcxplemdvbinom 41057 binomcxplemnotnn0 41060 |
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