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Mirrors > Home > MPE Home > Th. List > Mathboxes > binomcxplemcvg | Structured version Visualization version GIF version |
Description: Lemma for binomcxp 42627. The sum in binomcxplemnn0 42619 and its derivative (see the next theorem, binomcxplemdvsum 42625) 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 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐶 ∈ ℂ) |
4 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝑗 ∈ ℕ0) | |
5 | 3, 4 | bcccl 42609 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐶C𝑐𝑗) ∈ ℂ) |
6 | binomcxplem.f | . . . . 5 ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗)) | |
7 | 5, 6 | fmptd 7062 | . . . 4 ⊢ (𝜑 → 𝐹:ℕ0⟶ℂ) |
8 | 7 | adantr 481 | . . 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 2829 | . . . . . 6 ⊢ (𝐽 ∈ 𝐷 ↔ 𝐽 ∈ (◡abs “ (0[,)𝑅))) |
12 | absf 15222 | . . . . . . 7 ⊢ abs:ℂ⟶ℝ | |
13 | ffn 6668 | . . . . . . 7 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ) | |
14 | elpreima 7008 | . . . . . . 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 274 | . . . . 5 ⊢ (𝐽 ∈ 𝐷 ↔ (𝐽 ∈ ℂ ∧ (abs‘𝐽) ∈ (0[,)𝑅))) |
17 | 16 | simplbi 498 | . . . 4 ⊢ (𝐽 ∈ 𝐷 → 𝐽 ∈ ℂ) |
18 | 17 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → 𝐽 ∈ ℂ) |
19 | 16 | simprbi 497 | . . . . 5 ⊢ (𝐽 ∈ 𝐷 → (abs‘𝐽) ∈ (0[,)𝑅)) |
20 | 0re 11157 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
21 | ssrab2 4037 | . . . . . . . . . 10 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ } ⊆ ℝ | |
22 | ressxr 11199 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
23 | 21, 22 | sstri 3953 | . . . . . . . . 9 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ } ⊆ ℝ* |
24 | supxrcl 13234 | . . . . . . . . 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 13328 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → ((abs‘𝐽) ∈ (0[,)𝑅) ↔ ((abs‘𝐽) ∈ ℝ ∧ 0 ≤ (abs‘𝐽) ∧ (abs‘𝐽) < 𝑅))) | |
28 | 20, 26, 27 | mp2an 690 | . . . . . 6 ⊢ ((abs‘𝐽) ∈ (0[,)𝑅) ↔ ((abs‘𝐽) ∈ ℝ ∧ 0 ≤ (abs‘𝐽) ∧ (abs‘𝐽) < 𝑅)) |
29 | 28 | simp3bi 1147 | . . . . 5 ⊢ ((abs‘𝐽) ∈ (0[,)𝑅) → (abs‘𝐽) < 𝑅) |
30 | 19, 29 | syl 17 | . . . 4 ⊢ (𝐽 ∈ 𝐷 → (abs‘𝐽) < 𝑅) |
31 | 30 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → (abs‘𝐽) < 𝑅) |
32 | 1, 8, 9, 18, 31 | radcnvlt2 25778 | . 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 767 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐽 ∈ ℂ) ∧ 𝑏 = 𝐽) ∧ 𝑘 ∈ ℕ) → 𝑏 = 𝐽) | |
36 | 35 | oveq1d 7372 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐽 ∈ ℂ) ∧ 𝑏 = 𝐽) ∧ 𝑘 ∈ ℕ) → (𝑏↑(𝑘 − 1)) = (𝐽↑(𝑘 − 1))) |
37 | 36 | oveq2d 7373 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝐽 ∈ ℂ) ∧ 𝑏 = 𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1))) = ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1)))) |
38 | 37 | mpteq2dva 5205 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ ℂ) ∧ 𝑏 = 𝐽) → (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1)))) = (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1))))) |
39 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℂ) → 𝐽 ∈ ℂ) | |
40 | nnex 12159 | . . . . . . . 8 ⊢ ℕ ∈ V | |
41 | 40 | mptex 7173 | . . . . . . 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 6955 | . . . . 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 13914 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → seq1( + , (𝐸‘𝐽)) = seq1( + , (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1)))))) |
46 | eqid 2736 | . . . 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 42617 | . . 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 512 | 1 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → (seq0( + , (𝑆‘𝐽)) ∈ dom ⇝ ∧ seq1( + , (𝐸‘𝐽)) ∈ dom ⇝ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3407 Vcvv 3445 ⊆ wss 3910 class class class wbr 5105 ↦ cmpt 5188 ◡ccnv 5632 dom cdm 5633 “ cima 5636 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 supcsup 9376 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 ℝ*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 ℕcn 12153 ℕ0cn0 12413 ℝ+crp 12915 [,)cico 13266 seqcseq 13906 ↑cexp 13967 abscabs 15119 ⇝ cli 15366 C𝑐cbcc 42606 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-fac 14174 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-prod 15789 df-fallfac 15890 df-bcc 42607 |
This theorem is referenced by: binomcxplemnotnn0 42626 |
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