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Mirrors > Home > MPE Home > Th. List > Mathboxes > binomcxplemradcnv | Structured version Visualization version GIF version |
Description: Lemma for binomcxp 42873. By binomcxplemfrat 42867 and radcnvrat 42830 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 483 | . . . . . . . . 9 ⊢ ((𝑏 = 𝑥 ∧ 𝑘 ∈ ℕ0) → 𝑏 = 𝑥) | |
3 | 2 | oveq1d 7405 | . . . . . . . 8 ⊢ ((𝑏 = 𝑥 ∧ 𝑘 ∈ ℕ0) → (𝑏↑𝑘) = (𝑥↑𝑘)) |
4 | 3 | oveq2d 7406 | . . . . . . 7 ⊢ ((𝑏 = 𝑥 ∧ 𝑘 ∈ ℕ0) → ((𝐹‘𝑘) · (𝑏↑𝑘)) = ((𝐹‘𝑘) · (𝑥↑𝑘))) |
5 | 4 | mpteq2dva 5238 | . . . . . 6 ⊢ (𝑏 = 𝑥 → (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))) = (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑥↑𝑘)))) |
6 | fveq2 6875 | . . . . . . . 8 ⊢ (𝑘 = 𝑦 → (𝐹‘𝑘) = (𝐹‘𝑦)) | |
7 | oveq2 7398 | . . . . . . . 8 ⊢ (𝑘 = 𝑦 → (𝑥↑𝑘) = (𝑥↑𝑦)) | |
8 | 6, 7 | oveq12d 7408 | . . . . . . 7 ⊢ (𝑘 = 𝑦 → ((𝐹‘𝑘) · (𝑥↑𝑘)) = ((𝐹‘𝑦) · (𝑥↑𝑦))) |
9 | 8 | cbvmptv 5251 | . . . . . 6 ⊢ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑥↑𝑘))) = (𝑦 ∈ ℕ0 ↦ ((𝐹‘𝑦) · (𝑥↑𝑦))) |
10 | 5, 9 | eqtrdi 2787 | . . . . 5 ⊢ (𝑏 = 𝑥 → (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘))) = (𝑦 ∈ ℕ0 ↦ ((𝐹‘𝑦) · (𝑥↑𝑦)))) |
11 | 10 | cbvmptv 5251 | . . . 4 ⊢ (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘)))) = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℕ0 ↦ ((𝐹‘𝑦) · (𝑥↑𝑦)))) |
12 | 1, 11 | eqtri 2759 | . . 3 ⊢ 𝑆 = (𝑥 ∈ ℂ ↦ (𝑦 ∈ ℕ0 ↦ ((𝐹‘𝑦) · (𝑥↑𝑦)))) |
13 | binomcxp.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
14 | 13 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) → 𝐶 ∈ ℂ) |
15 | simpr 485 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) → 𝑗 ∈ ℕ0) | |
16 | 14, 15 | bcccl 42855 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑗 ∈ ℕ0) → (𝐶C𝑐𝑗) ∈ ℂ) |
17 | binomcxplem.f | . . . 4 ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗)) | |
18 | 16, 17 | fmptd 7095 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → 𝐹:ℕ0⟶ℂ) |
19 | binomcxplem.r | . . 3 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) | |
20 | fvoveq1 7413 | . . . . . 6 ⊢ (𝑘 = 𝑖 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑖 + 1))) | |
21 | fveq2 6875 | . . . . . 6 ⊢ (𝑘 = 𝑖 → (𝐹‘𝑘) = (𝐹‘𝑖)) | |
22 | 20, 21 | oveq12d 7408 | . . . . 5 ⊢ (𝑘 = 𝑖 → ((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘)) = ((𝐹‘(𝑖 + 1)) / (𝐹‘𝑖))) |
23 | 22 | fveq2d 6879 | . . . 4 ⊢ (𝑘 = 𝑖 → (abs‘((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘))) = (abs‘((𝐹‘(𝑖 + 1)) / (𝐹‘𝑖)))) |
24 | 23 | cbvmptv 5251 | . . 3 ⊢ (𝑘 ∈ ℕ0 ↦ (abs‘((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘)))) = (𝑖 ∈ ℕ0 ↦ (abs‘((𝐹‘(𝑖 + 1)) / (𝐹‘𝑖)))) |
25 | nn0uz 12843 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
26 | 0nn0 12466 | . . . 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 485 | . . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 = 𝑖) → 𝑗 = 𝑖) | |
30 | 29 | oveq2d 7406 | . . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) ∧ 𝑗 = 𝑖) → (𝐶C𝑐𝑗) = (𝐶C𝑐𝑖)) |
31 | simpr 485 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0) | |
32 | ovexd 7425 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → (𝐶C𝑐𝑖) ∈ V) | |
33 | 28, 30, 31, 32 | fvmptd 6988 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → (𝐹‘𝑖) = (𝐶C𝑐𝑖)) |
34 | elfznn0 13573 | . . . . . . 7 ⊢ (𝐶 ∈ (0...(𝑖 − 1)) → 𝐶 ∈ ℕ0) | |
35 | 34 | con3i 154 | . . . . . 6 ⊢ (¬ 𝐶 ∈ ℕ0 → ¬ 𝐶 ∈ (0...(𝑖 − 1))) |
36 | 35 | ad2antlr 725 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → ¬ 𝐶 ∈ (0...(𝑖 − 1))) |
37 | 13 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐶 ∈ ℂ) |
38 | simpr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0) | |
39 | 37, 38 | bcc0 42856 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝐶C𝑐𝑖) = 0 ↔ 𝐶 ∈ (0...(𝑖 − 1)))) |
40 | 39 | necon3abid 2976 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝐶C𝑐𝑖) ≠ 0 ↔ ¬ 𝐶 ∈ (0...(𝑖 − 1)))) |
41 | 40 | adantlr 713 | . . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → ((𝐶C𝑐𝑖) ≠ 0 ↔ ¬ 𝐶 ∈ (0...(𝑖 − 1)))) |
42 | 36, 41 | mpbird 256 | . . . 4 ⊢ (((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) ∧ 𝑖 ∈ ℕ0) → (𝐶C𝑐𝑖) ≠ 0) |
43 | 33, 42 | eqnetrd 3007 | . . 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 42867 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → (𝑘 ∈ ℕ0 ↦ (abs‘((𝐹‘(𝑘 + 1)) / (𝐹‘𝑘)))) ⇝ 1) |
48 | ax-1ne0 11158 | . . . 4 ⊢ 1 ≠ 0 | |
49 | 48 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → 1 ≠ 0) |
50 | 12, 18, 19, 24, 25, 27, 43, 47, 49 | radcnvrat 42830 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → 𝑅 = (1 / 1)) |
51 | 1div1e1 11883 | . 2 ⊢ (1 / 1) = 1 | |
52 | 50, 51 | eqtrdi 2787 | 1 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ ℕ0) → 𝑅 = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 {crab 3429 Vcvv 3470 class class class wbr 5138 ↦ cmpt 5221 dom cdm 5666 ‘cfv 6529 (class class class)co 7390 supcsup 9414 ℂcc 11087 ℝcr 11088 0cc0 11089 1c1 11090 + caddc 11092 · cmul 11094 ℝ*cxr 11226 < clt 11227 − cmin 11423 / cdiv 11850 ℕ0cn0 12451 ℝ+crp 12953 ...cfz 13463 seqcseq 13945 ↑cexp 14006 abscabs 15160 ⇝ cli 15407 C𝑐cbcc 42852 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-inf2 9615 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 ax-pre-sup 11167 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-isom 6538 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7650 df-om 7836 df-1st 7954 df-2nd 7955 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-1o 8445 df-er 8683 df-pm 8803 df-en 8920 df-dom 8921 df-sdom 8922 df-fin 8923 df-sup 9416 df-inf 9417 df-oi 9484 df-card 9913 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-div 11851 df-nn 12192 df-2 12254 df-3 12255 df-n0 12452 df-z 12538 df-uz 12802 df-q 12912 df-rp 12954 df-ioo 13307 df-ico 13309 df-fz 13464 df-fzo 13607 df-fl 13736 df-seq 13946 df-exp 14007 df-fac 14213 df-hash 14270 df-shft 14993 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15612 df-prod 15829 df-fallfac 15930 df-bcc 42853 |
This theorem is referenced by: binomcxplemdvbinom 42869 binomcxplemnotnn0 42872 |
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