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Mirrors > Home > MPE Home > Th. List > Mathboxes > binomcxplemcvg | Structured version Visualization version GIF version |
Description: Lemma for binomcxp 44352. The sum in binomcxplemnn0 44344 and its derivative (see the next theorem, binomcxplemdvsum 44350) converge, as long as their base 𝐽 is within the disk of convergence. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (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*, < ) |
binomcxplem.e | ⊢ 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1))))) |
binomcxplem.d | ⊢ 𝐷 = (◡abs “ (0[,)𝑅)) |
Ref | Expression |
---|---|
binomcxplemcvg | ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → (seq0( + , (𝑆‘𝐽)) ∈ dom ⇝ ∧ seq1( + , (𝐸‘𝐽)) ∈ dom ⇝ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | binomcxplem.s | . . 3 ⊢ 𝑆 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ0 ↦ ((𝐹‘𝑘) · (𝑏↑𝑘)))) | |
2 | binomcxp.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
3 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐶 ∈ ℂ) |
4 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝑗 ∈ ℕ0) | |
5 | 3, 4 | bcccl 44334 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐶C𝑐𝑗) ∈ ℂ) |
6 | binomcxplem.f | . . . . 5 ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗)) | |
7 | 5, 6 | fmptd 7133 | . . . 4 ⊢ (𝜑 → 𝐹:ℕ0⟶ℂ) |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → 𝐹:ℕ0⟶ℂ) |
9 | binomcxplem.r | . . 3 ⊢ 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) | |
10 | binomcxplem.d | . . . . . . 7 ⊢ 𝐷 = (◡abs “ (0[,)𝑅)) | |
11 | 10 | eleq2i 2830 | . . . . . 6 ⊢ (𝐽 ∈ 𝐷 ↔ 𝐽 ∈ (◡abs “ (0[,)𝑅))) |
12 | absf 15372 | . . . . . . 7 ⊢ abs:ℂ⟶ℝ | |
13 | ffn 6736 | . . . . . . 7 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ) | |
14 | elpreima 7077 | . . . . . . 7 ⊢ (abs Fn ℂ → (𝐽 ∈ (◡abs “ (0[,)𝑅)) ↔ (𝐽 ∈ ℂ ∧ (abs‘𝐽) ∈ (0[,)𝑅)))) | |
15 | 12, 13, 14 | mp2b 10 | . . . . . 6 ⊢ (𝐽 ∈ (◡abs “ (0[,)𝑅)) ↔ (𝐽 ∈ ℂ ∧ (abs‘𝐽) ∈ (0[,)𝑅))) |
16 | 11, 15 | bitri 275 | . . . . 5 ⊢ (𝐽 ∈ 𝐷 ↔ (𝐽 ∈ ℂ ∧ (abs‘𝐽) ∈ (0[,)𝑅))) |
17 | 16 | simplbi 497 | . . . 4 ⊢ (𝐽 ∈ 𝐷 → 𝐽 ∈ ℂ) |
18 | 17 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → 𝐽 ∈ ℂ) |
19 | 16 | simprbi 496 | . . . . 5 ⊢ (𝐽 ∈ 𝐷 → (abs‘𝐽) ∈ (0[,)𝑅)) |
20 | 0re 11260 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
21 | ssrab2 4089 | . . . . . . . . . 10 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ } ⊆ ℝ | |
22 | ressxr 11302 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
23 | 21, 22 | sstri 4004 | . . . . . . . . 9 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ } ⊆ ℝ* |
24 | supxrcl 13353 | . . . . . . . . 9 ⊢ ({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ } ⊆ ℝ* → sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*) | |
25 | 23, 24 | ax-mp 5 | . . . . . . . 8 ⊢ sup({𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ* |
26 | 9, 25 | eqeltri 2834 | . . . . . . 7 ⊢ 𝑅 ∈ ℝ* |
27 | elico2 13447 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → ((abs‘𝐽) ∈ (0[,)𝑅) ↔ ((abs‘𝐽) ∈ ℝ ∧ 0 ≤ (abs‘𝐽) ∧ (abs‘𝐽) < 𝑅))) | |
28 | 20, 26, 27 | mp2an 692 | . . . . . 6 ⊢ ((abs‘𝐽) ∈ (0[,)𝑅) ↔ ((abs‘𝐽) ∈ ℝ ∧ 0 ≤ (abs‘𝐽) ∧ (abs‘𝐽) < 𝑅)) |
29 | 28 | simp3bi 1146 | . . . . 5 ⊢ ((abs‘𝐽) ∈ (0[,)𝑅) → (abs‘𝐽) < 𝑅) |
30 | 19, 29 | syl 17 | . . . 4 ⊢ (𝐽 ∈ 𝐷 → (abs‘𝐽) < 𝑅) |
31 | 30 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → (abs‘𝐽) < 𝑅) |
32 | 1, 8, 9, 18, 31 | radcnvlt2 26476 | . 2 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → seq0( + , (𝑆‘𝐽)) ∈ dom ⇝ ) |
33 | binomcxplem.e | . . . . . . 7 ⊢ 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1))))) | |
34 | 33 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℂ) → 𝐸 = (𝑏 ∈ ℂ ↦ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))))) |
35 | simplr 769 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐽 ∈ ℂ) ∧ 𝑏 = 𝐽) ∧ 𝑘 ∈ ℕ) → 𝑏 = 𝐽) | |
36 | 35 | oveq1d 7445 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐽 ∈ ℂ) ∧ 𝑏 = 𝐽) ∧ 𝑘 ∈ ℕ) → (𝑏↑(𝑘 − 1)) = (𝐽↑(𝑘 − 1))) |
37 | 36 | oveq2d 7446 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝐽 ∈ ℂ) ∧ 𝑏 = 𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1))) = ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1)))) |
38 | 37 | mpteq2dva 5247 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ ℂ) ∧ 𝑏 = 𝐽) → (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))) = (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1))))) |
39 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℂ) → 𝐽 ∈ ℂ) | |
40 | nnex 12269 | . . . . . . . 8 ⊢ ℕ ∈ V | |
41 | 40 | mptex 7242 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1)))) ∈ V |
42 | 41 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℂ) → (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1)))) ∈ V) |
43 | 34, 38, 39, 42 | fvmptd 7022 | . . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℂ) → (𝐸‘𝐽) = (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1))))) |
44 | 17, 43 | sylan2 593 | . . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → (𝐸‘𝐽) = (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1))))) |
45 | 44 | seqeq3d 14046 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → seq1( + , (𝐸‘𝐽)) = seq1( + , (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1)))))) |
46 | eqid 2734 | . . . 4 ⊢ (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1)))) = (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1)))) | |
47 | 1, 9, 46, 8, 18, 31 | dvradcnv2 44342 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → seq1( + , (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1))))) ∈ dom ⇝ ) |
48 | 45, 47 | eqeltrd 2838 | . 2 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → seq1( + , (𝐸‘𝐽)) ∈ dom ⇝ ) |
49 | 32, 48 | jca 511 | 1 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → (seq0( + , (𝑆‘𝐽)) ∈ dom ⇝ ∧ seq1( + , (𝐸‘𝐽)) ∈ dom ⇝ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 {crab 3432 Vcvv 3477 ⊆ wss 3962 class class class wbr 5147 ↦ cmpt 5230 ◡ccnv 5687 dom cdm 5688 “ cima 5691 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 supcsup 9477 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 − cmin 11489 ℕcn 12263 ℕ0cn0 12523 ℝ+crp 13031 [,)cico 13385 seqcseq 14038 ↑cexp 14098 abscabs 15269 ⇝ cli 15516 C𝑐cbcc 44331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-fac 14309 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-prod 15936 df-fallfac 16039 df-bcc 44332 |
This theorem is referenced by: binomcxplemnotnn0 44351 |
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