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| Mirrors > Home > MPE Home > Th. List > Mathboxes > binomcxplemcvg | Structured version Visualization version GIF version | ||
| Description: Lemma for binomcxp 44376. The sum in binomcxplemnn0 44368 and its derivative (see the next theorem, binomcxplemdvsum 44374) 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 44358 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐶C𝑐𝑗) ∈ ℂ) |
| 6 | binomcxplem.f | . . . . 5 ⊢ 𝐹 = (𝑗 ∈ ℕ0 ↦ (𝐶C𝑐𝑗)) | |
| 7 | 5, 6 | fmptd 7134 | . . . 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 2833 | . . . . . 6 ⊢ (𝐽 ∈ 𝐷 ↔ 𝐽 ∈ (◡abs “ (0[,)𝑅))) |
| 12 | absf 15376 | . . . . . . 7 ⊢ abs:ℂ⟶ℝ | |
| 13 | ffn 6736 | . . . . . . 7 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ) | |
| 14 | elpreima 7078 | . . . . . . 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 11263 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
| 21 | ssrab2 4080 | . . . . . . . . . 10 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ } ⊆ ℝ | |
| 22 | ressxr 11305 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
| 23 | 21, 22 | sstri 3993 | . . . . . . . . 9 ⊢ {𝑟 ∈ ℝ ∣ seq0( + , (𝑆‘𝑟)) ∈ dom ⇝ } ⊆ ℝ* |
| 24 | supxrcl 13357 | . . . . . . . . 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 2837 | . . . . . . 7 ⊢ 𝑅 ∈ ℝ* |
| 27 | elico2 13451 | . . . . . . 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 1148 | . . . . 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 26462 | . 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 7446 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐽 ∈ ℂ) ∧ 𝑏 = 𝐽) ∧ 𝑘 ∈ ℕ) → (𝑏↑(𝑘 − 1)) = (𝐽↑(𝑘 − 1))) |
| 37 | 36 | oveq2d 7447 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝐽 ∈ ℂ) ∧ 𝑏 = 𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑘 · (𝐹‘𝑘)) · (𝑏↑(𝑘 − 1))) = ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1)))) |
| 38 | 37 | mpteq2dva 5242 | . . . . . 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 12272 | . . . . . . . 8 ⊢ ℕ ∈ V | |
| 41 | 40 | mptex 7243 | . . . . . . 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 7023 | . . . . 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 14050 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → seq1( + , (𝐸‘𝐽)) = seq1( + , (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1)))))) |
| 46 | eqid 2737 | . . . 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 44366 | . . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ 𝐷) → seq1( + , (𝑘 ∈ ℕ ↦ ((𝑘 · (𝐹‘𝑘)) · (𝐽↑(𝑘 − 1))))) ∈ dom ⇝ ) |
| 48 | 45, 47 | eqeltrd 2841 | . 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 1087 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ◡ccnv 5684 dom cdm 5685 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supcsup 9480 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 ℕcn 12266 ℕ0cn0 12526 ℝ+crp 13034 [,)cico 13389 seqcseq 14042 ↑cexp 14102 abscabs 15273 ⇝ cli 15520 C𝑐cbcc 44355 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-fac 14313 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-prod 15940 df-fallfac 16043 df-bcc 44356 |
| This theorem is referenced by: binomcxplemnotnn0 44375 |
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